Act 1. Shadowheart is attracted to your Tav(You can read her thoughts at some points), but is not in love yet and doesn't really know what she wants. Flirts with other companions, shows some interest in Karlach. Doesn't act on anything though, doesn't sleep with others if not romanced. Invites Tav at Tiefling party to romantic evening(Option to get her say not today if ask her about shared arrangment. You can read it differently, sure. I read it more as she points that she wants to spend actually not in friend's company, but romantic one with Tav. Tav is playing dumb obviously. Also it does not end up in sex or anything.).
Act 2. She actually falls in love but is hindered by her faith and duty(Thats the key here). Denies any poly proposal at this point(Even with her conflict between faith/love. Isn't it a bit selfish or hypocritical if she was actually poly all this time?), she wants Tav all for herself. Post Shadowfell she actually goes on confession about this and tells Tav first time in the game that she wants to be with Tav now and always.
Act 3. She actually is ok with open relationship if Tav is. She is ok at "having fun" with drows, she is ok with Halsin fling(Again, I consider this as fling, because of her absolute 180 turn with act 2 behaviour), ok with Mizora. Never asks Tav about opening relationship herself and stays faithful till the end(So its only on Tav to "not be faithful").
She is monogamous, but ok with opening relationship(Which makes sense considering how she was raised and lived her life pre Tav).
Thats how I read it.