they have a tape measure and they go around just below where your bra line should be. and then they go up top to measure around to see cup size. and that's how they do it. they substract the smaller number from the bigger number to figure out cup size and the measurement where your bra line should be is the waist like like 32, 34, 36 etc.
It's not embarrassing at all. They are trained professionals, and aren't going to judge you, or whatever it is you're worried about. I recommend Victoria's Secret, personally. They actually have bra measurement specialists at most of their stores, I believe, and my experiences with them have been great. Basically all they do is measure around your t*ts with a tape measure a couple of times, and then you're done. It's simple.
You can get measured at almost any store selling bras. JCPenny's, Victoria Secret's, even Walmart. It's a little uncomfortable, but no big deal. You leave your bra on and they measure you with a measuring tape. It's definitely worth it as well. The right bra can not only make you feel more comfortable, but also make your breasts look more attractive. Don't really think about the embarrassing side of things. It's over in just in few seconds.
There's really nothing to be embarrassed about. The people who do the measurements are trained, and they measure women all the time, so they're not going to think anything of you for asking. It doesn't even take that long. They just put the tape measure around you a couple of times and tell you your bra size. I think the benefits of having a bra that actually fits outweigh any awkwardness.
I go to a special shop focused on larger sizes and I love going there =) the women are so nice and funny and they bring you all your bras and what they think you might need/like. ha and they are dead honest and try their best to get the perfect fit. The shop's called your perfect fit haha. And I'm actually so comfortable there with them that You just change right there so they don't have to keep popping in and out of the room
It's totally nonchalant, so don't feel embarrassed. Once they measure you they will bring you tons of pretty bras in your size. It's kind of relaxing... I like bra shopping.