Bi-directional replication for the same MySQL table
I'm assuming these are different instances of MySQL (most likely on different servers).

You can set up Master-master replication, where dbA is a master of dbB and vice versa.

Then, to avoid auto-increment collisions, set the following in my.cnf on dbA:

auto_increment_increment=2
auto_increment_offset=1


And the following on dbB:

auto_increment_increment=2
auto_increment_offset=2