I had a doula for my birth (a home birth, so slightly different experience than most). To be honest, it wasn't the best experience only because I didn't really get in the prenatal visits or have a chance to build up that trust. She was mostly in the way. She went away on holiday when she should have already been on-call for us and she barely made it from the airport to my birth in time for me to start pushing. That said, I generally think doulas are fantastic (my best friend is a doula and she's awesome! sadly, she lives in the U.S. and I'm in the UK now). But I do think it's really, really important to choose someone you have a bond with and to take the time to build up that trust during your prenatal visits. I had my doubts about hiring the person we did just because I didn't feel a real bond with her, but she was one of only two doulas working in my area who had any experience with home birth, so it was slim pickings. So it was basically her or no one at all, and I really wanted to make sure I had all the support I could since it was my first birth. Two to three visits (you can also always have more than that, you might just have to pay extra) should be enough to build up that trust with the right person though.
As for having someone there that you aren't familiar with, no it's not weird. I had never met one of the midwives who attended my birth so that was a bit weird (I think at one point I told her she was getting me off my game and she had to leave the room! lol), but having my doula there (even when I wasn't totally pleased with her) wasn't weird or uncomfortable. At that point you totally don't care who sees you naked or whatnot. You're in your own place and doing your own thing. The only thing I would think about is if you're someone who doesn't want a whole lot of people in the room in general, then a doula might just seem like an extra person. That's not a reason not to have one, but you might need to be very specific about asking people to leave if you feel a bit crowded in. I know I did earlier in my labour, though later on, I really didn't notice as much.
We paid £200 (about $350) for 2 prenatal visits (should have been 3), attendance at the birth (though she only made it for the last 4 hours) and one postnatal visit (though I think she would have done 2 if we wanted). She was still in training, which was why it was so cheap. The other doulas we interviewed charged £550.
If I had it to do next time, I don't know that I'd hire a doula unless I knew I'd have to give birth in hospital. Like I said, I think doulas are fantastic! My friend is one and she's awesome and devoted to the women she works with. But having done it mostly on my own (it was just me and my husband until I called the midwives and they arrived when I was already pushing, so I didn't actually use or need much other support), I think I'd probably prefer to do that again assuming I could have another home birth. But if I knew I would have to give birth in hospital or had any known complications, I think I would appreciate the extra support. Also, something I think I might consider doing next time is just hiring a postnatal doula (they only come to support you after the birth and help with breastfeeding, debriefing you on your birth experience, provide some childcare for older children, help with basic chores, etc.). The first few weeks are hard work and exhausting and I can see the benefit of having extra support then.
So this sounds a bit negative, but really doulas are great and I would definitely look into hiring one, especially if you are birthing in hospital and really want someone to act as your advocate and protect your birthing space and plan. I was lucky to have midwives who were experienced and in support of home/natural birth and to have a really supportive husband, plus a really straightforward, calm birth. I think I would have appreciated having someone there to provide additional support if I hadn't had it this way.