What I remember from my case few years ago, incoming SMS were handled to both SIMs. Outgoing SMS from my number didn't appear on second one (so shared SMS history isn't complete in that sens). I didn't use MMS. I think all those behaviors can be different, depending on provider config.

Okay, that is a problem then. If both phones cannot see all messages, how would both parties have full context on the conversation? This would inevitably lead to both of us sending a response.