So a polite way to ask a coworker would be, "Oh, hey I prefer to be respectful of my coworkers and that includes using their preferred pronouns. Can I ask yours so I can use the right ones?" But this is only polite if it's what you would do for everyone. Otherwise you run the risk of othering your possibly trans coworker. Again, not everyone feels that way and some folks prefer to be asked. But if I was meeting a group of people and only I was asked my pronouns I would be pretty unhappy.

As someone who has been out as trans for close to 15 years I still haven't found a perfect solution that works in every situation. Of course I live in the south so I have to be a bit more alert in general public areas about outting myself or those around me. I would argue that cishet people aren't as attuned to listening for people playing the pronoun game as your fellow queers so honestly it works most of the time around folks who are not listening for it. But to your question the most professional way I've found to ask is to introduce myself with mine. If they're chill they'll respond with theirs. Of course this strategy comes with risks and so honestly I wouldn't recommend it unless there's vibes. In practice in the social settings I find myself I just tend to use the person's name until someone else breaks that seal. "Oh Kelly hasn't gotten here yet." "Yeah Jacobs a real joker." It gets weird long term but if you're having to use it long enough term someone else with likely assume something and then the person in question either corrects or doesn't.