What happens to the information on the event horizons of two merging black holes?

Not every scientist agrees that information is "encoded" on the surface of a black hole. Many scientists believe black holes actually destroy information. In fact, Stephen Hawking and Kip Thorne made a famous wager against John Preskill about whether information is destroyed by black hole.

The simplest (and in my opinion, most likely) answer to your question is that black hole event horizons don't encode any information at all. The black destroys it. Which is why we say "black holes have no hair". Once you make a black hole out of any material, you can no longer tell what went into it. If you make one of photons or neutrinos or neutrons or whatever, all you know after the black hole forms is the amount of mass/energy it contains.

So when two black holes collide, their event horizons still contain zero information. Zero from the first black hole plus zero from the second.