I hope I'm not repeating anything anyone else said!
The way to ask someone to get tested for HIV or STD is to say during a regular conversation, O by the way, when is the last time you were tested? How did it work out? Have you slept with onyone since then? Sure, people LIE But I would not sleep with someone, or allow my husband to sleep with someone, who could not have this conversation. If anyone is too nervous or immature to talk openly about their status- It's a huge red flag.
I don't know how different my relationship is: I would say our only rule is being honest. We know everything. I complain/talk about my lovers to my husband and vice versa. But I would never pick or veto someone he could date... I feel like it isn't my business (although there are some exceptions as described above). We have separate bedrooms. We are both allowed to have people over when the other (and our child) isn't home. I encourage my husband to invite his girlfriend to spend the night when I'm out of town, and if our son is out of town, he'll stay with her. A primary element is our son as he doesn't know (exactly and we have to nogiatoe child care and chores). My husband has never met my current love and may never. My husband gf and I are not close but on good terms (she has dinner here every few weeks).
The only other rules we have is no sex parities for my husband. Because two of my friends were sex addicts, its too much for me to berar, although he does do s/mplay parties.
Other than that, it's nice, Kind of like best of bother worlds, we have our own space and activities and lovers, but we work domestically well together and our good friends. I do miss having an intense romantic partner, but that might not be in the cards for me. I also don't consider myself poly because I primarily want freedom!! Not two husbands!