This is my approach. You are gonna get a kind of Americanized version because I can’t help it, but:

We have an Irinya in our department, soft r, I sounds like our long E - I am not gonna call her Irine! Soft R I am familiar with from Spanish, I’m sure it’s not spot on as she is Eastern European but close enough for her, she says it sounds right. One lady on our team can’t even hear what the R sounds like and can’t make the sound, she calls her eeReenya with the hard R. Still acceptable to Irinya, that’s what matters.

If it’s some sounds I can’t pronounce, I’d ask what they want to be called if I can’t make those sounds but in general yes I try to pronounce people’s names as they do, to the extent I can.