Its not so much "processors" talking to each other as caches.

The basic theory is called "MESI": Modified / Exclusive / Shared / Invalid. When a remote core reads a cache-line, if no one else has that cache-line then its the Exclusive owner. But if multiple caches have the cache-line, then it is a Shared cache line instead. If a remote cache changes, then your current cache-line becomes Invalid, while the cache line that's valid is placed into the Modified state (cache is correct, RAM is incorrect).

MESI isn't how things are done in practice: additional messages and states are piled on top in proprietary ways. But MESI will get you to the "basic textbook" concept at least.

EDIT: Oh, and "snooping". Caches "snoop" on each other's communications, which helps keep the MESI state up to date. Now you can have directory-based snooping (kind of a centralized push), or rings, or whatever. Those details are proprietary and change from chip to chip. In practice, these details are almost never published, so your best bet to understanding is "Something MESI-like is going on..."