Following up on a postdoc application at an academic conference in person I will graduate with my PhD in a few months and I have starting applying for postdoc positions at national research labs and universities. Our field is relatively small (a few hundred people) and most of us know each other and the rest we know by name or by face. My question is that, after I apply for a postdoc position, I then meet the hiring scientist at a conference, then how appropriate is it ask them about my application? If it is appropriate, then what is the best way to do so? In person? In email, this week? In email, after we return? Or something else?

This is in the field of hard sciences in the USA. One potential employer is a friend I have known since my beginning graduate school. Another potential employer, I just know her by name and never had any contact of any kind besides my application. Should I ask/say anything? I am obviously curious how's the process going and if I am high on the list. But what would I say? Or is it better to just be friendly but professional and pretend like I didn't apply and not mention anything at all? I can only imagine a useless "we are still looking and evaluating applications". Is any other response even possible if I ask the right questions in the right way?

It has been about a month since I applied. And yes I asked my advisor but he wasn't helpful. Both solicitations were clear that they wanted the postdoc to start as soon as possible (three months) so a month seems like a fair amount of time to follow-up. And after the application acknowledgement I haven't heard anything from either scientist.
You could try to reach out by email and tell them that you've applied for a postdoc position in their lab, and that you are going to be at the conference where you'd love a chance to discuss your research contributions and qualifications as well as the opportunities that they may have. If they have an interest and tell you as much, by the time you two meet you both know why you're talking to each other.