2-way means the routers are available to exchange the LSDB (Link State Database). This means these routers can now start sending each other database updates. Once the databases are the same the routers will move into a Full state. Meaning both routers LSDB are the same.
OSPF has designated routers and backup designated routers, however these are not elected on point-to-point links because they are meaningless on such connections.