Happened (sumbaino) means to happen, with implication of occurring in connection w other events. The perfect tense treats what is certain to befall as already accomplished.

True (alethes) means conforming to reality, unconcealed, manifest or in accordance with fact. Peter is stamping this proverb as in accord with reality.

Proverb (paroimia from pará = by + oímos = a way, a highway) is literally something "by the way", a byword which is a short saying illustrating a general principle. It describes a pithy maxim giving expression to some observed event whose content has allegorical import. It also describes a brief communication containing truths designed for initiates and as such can be a veiled saying in which especially lofty ideas are concealed (used especially this way by John)

The fact that "proverb" is singular indicates that the message conveyed by the two examples is essentially the same, namely both point out examples of repulsive actions and to the character that these actions reveal. Simply put dogs will act like dogs thus showing that they were dogs all along and the same for pigs. The false actions of the teachers reflect who they really have been all along - false teachers. Furthermore this proverb was well known among the rabbis and the actions described was well known among the pagans.

A DOG RETURNS TO ITS OWN VOMIT: kuon epistrepsas (AAPMSN) epi to idion exerama, (Pr26:11)

In Peter's day dogs were not pampered pets like they are today! The Jews in fact spoke of the hated Gentiles as “dogs” because a dog was nothing but a filthy scavenger who ran in packs and lived on garbage!  Peter chooses a word which is the epitome of disrespect and revulsion to describe these false teachers who have known the truth but have turned away from it.

Returns (epistrepho) return to a point or area where one has been before, in this case to their own regurgitated gastric contents.

Vomit (exerama) means that which is thrown out (vomited). The picture here is not just of a dog "sniffing" the vomitus, but of actually lapping up what had been disgorged. Both dogs and pigs were considered vile by the Jews. The action of the dog (and the pig) reveals its true nature. The false teachers appeared to be renewed persons, having made professions of faith, but their false teaching and persistent practice revealed that they were still lost apostates.

Matthew Henry describes them as those who

have licked up their own vomit again, returning to the same errors and impieties that they had once cast off and seemed to detest and loathe, and wallowing in that filthiness from which they appeared once to be really cleansed.

AND A SOW AFTER WASHING...TO WALLOWING IN THE MIRE: kai: us lousamene (AMPFSN) eis kulismon borborou:

Washing (luo) means to bathe oneself and specifically refers to washing the whole body and not part of it. The middle voice indicates that the washing was not forcibly applied against the pig's will but that the pig the washing himself. It pictures however an outer cleansing not an internal cleansing. The clear implication is that these false teachers had been fully exposed to the genuine gospel yet internally remained "full of dead men's bones". By the way don't press the proverb too far. Peter is not alluding to washing as a picture of baptism. The picture implies an outward cleansing form "defiling conduct". But the outward cleansing was readily undone by the pig's innate natural urge to find "relief" by returning to the mire. A pig can stay clean only a short time and then must head for the nearest mud hole. We do not condemn a pig for acting like a pig because it has a pig’s nature. If we saw a sheep heading for the mire, we would be concerned!

Peter’s point is that mere religious profession or even outward change does not change a person’s heart. These “professors but not possessors” seemed to experience salvation, but in due time they drifted back to the life that suited their nature. Certainly the dog feels better after emptying his stomach, but it is still a dog. “Having an experience” did not change his nature. Quite the contrary, it only gave further evidence of his “dog nature,” because he came back and (just like a dog) lapped up his own vomit. It is a disgusting picture, but that is exactly the response Peter wanted to produce. The apostasy of the false teachers reveals their true disgusting nature.

The principle brought out by Peter calls for us all to apply this truth to our lives and reflect on our choices. Our choices are seen to be consistent with what we are. A good tree bears good fruit, a brackish spring pours forth brackish water, and the pig returns to wallowing in the mire. What do my choices today & this past week, this past year, etc loudly proclaim about who I really am? For heaven's sake, we need to be honest with ourselves! The dog and pig in context picture temporary external change resulting from conformity to a false profession of faith much like a chameleon blends with its surroundings whatever they might be. True faith is fruitful faith. False "faith" is shown by absence of good fruit (cf Peter's earlier comments 2Pe 1:8-10) Ignorance in the spiritual realm is not bliss but leads to fleshly indulgence. Unsaved people lack spiritual intelligence (Ho4:6), and this causes them to give themselves to all kinds of fleshly and worldly indulgences (Ac17:30; Ep4:17ff). Since we were born with a fallen nature it is natural for us to live sinful lives. Nature determines appetites and actions. A dog and a pig behave differently because they have different natures.

Jesus also used the designations "dogs" and "swine" in speaking of those opposed to God and his Word (cf Mt 7:6). The underlying principle of both examples is the same: these apostates (whether false teachers, their victims, or both) never were what they seemed to be and returned to what they had been all along. Dogs and pigs can be scrubbed but not kept clean, for it is in their very nature to return to unclean living. Such apostates are in a tighter bondage, they are farther from the truth, and they are deeper in spiritual filth than ever before. These individuals have a religious "profession" or outward show without a regenerating inner change that affects his true nature. Since such a person will revert to their true nature, perseverance in the faith to the end is an important criterion of a genuine change in one's nature

MacDonald reminds us that

This passage should not be used to teach that true believers may fall from grace and be lost. These people never were true believers. They never received a new nature. They demonstrated by their last state that their nature was still unclean and evil. The lesson is, of course, that reformation alone is not only insufficient, but is positively dangerous, because it can lull a person into a false security. Man can receive a new nature only by being born again. He is born again through repentance toward God and faith in our Lord Jesus Christ.

Warren Wiersbe summarizes the characteristics of these false teachers:

"We may detect them by their exaltation of themselves instead of Christ; their counterfeit talk and “great swelling words”; their emphasis on making money; their great claims that they can change people; and their hidden lives of lust and sin. For the time being we cannot stop them except by teaching the Word sincerely, but one day God will expose them and judge them...(and Dr. Wiersbe goes on to add that...) "Now we can appreciate Peter’s admonition in 2Pe 1:10

Therefore, brethren, be all the more diligent to make certain about His calling and choosing you; for as long as you practice these things, you will never stumble” In other words, “Has your spiritual experience been genuine?” It is a startling fact that there are many people in our churches who have never truly been born again, but who are convinced that they are saved and going to heaven! They have had “an experience,” and perhaps look better (like the sow) and feel better (like the dog), but they have not been made better as “partakers of the divine nature.

Perhaps Peter recalled Judas, one of the Twelve, who was a tool of the devil and was never born again. Up to the very end, the other disciples did not know the truth about Judas and thought he was a spiritual man!" (bolding & coloring adding)

Kenneth Gangel in the Bible Knowledge Commentary has some sound advice for all saints in these last, deceptive, difficult days

"Believers today do well to heed Peter’s warning against false teachers, to learn how to discern truth for themselves, and to teach it to others. The false teachers will themselves meet destruction and others will be destroyed by them. But Christians can wage spiritual warfare more effectively if they know their spiritual enemies, the techniques that heretics use, and the end result of their deception." (bolding added) is, in itself, not similar) and we do not think and speak falsely but rightly when we describe the relationship as one of similarity."

J Vernon McGee summarizes this section in his unique pithy style