You can have in-game conversations with each other's characters at particular points of storyline (for example one comes up in the Ruined Temple, which is fairly early on) and you can vote on answers in some of the conversations. However this is currently an afterthought, you are right. There is also no way to romance each other's original characters, only the companions - AND in multiplayer, the second player doesn't get reputation updates with the companions, either. BUT this is very likely an Early Access issue/bug and something that might very well get resolved in Patch 5 which is next week. Hang in there!
My husband and I "solve" this problem by controlling each other's romance companion for each other's playthroughs. One of us still has the multiplayer "lead" in the session, and gets the save files etc, but he will log on and control my Astarion (whom I romance), and I play his Shadowheart (his romance) for his game. It's not exactly a solution but it works for us.
Although he does have to disconnect for me to be able to have pivotal camp conversations with Astarion, for example, otherwise the conversation won't trigger if the character is being controlled by someone at the time.
Let's see what Patch 5 brings in this regard.