Most likely you will keep just your knickers on, or they will give you disposable ones (my advice is to take some old ones of yours cos the disposable ones are gross!). They may ask you to completely strip but if you're not comfortable then just leave the knickers. You will lie on a towel so I wouldn't worry too much about leakage. I haven't had too many, but the couple of really good ones I've had I have to worry more about drooling as I begin to doze off! They'll show you to the room and then leave you to strip off, get on the table and get under the towels they have there. A good masseuse should have good "towel management" so that you never feel exposed, even if you're completely starkers underneath.
Not sure that you need to worry about what they're using as nothing is going to be anywhere too close to breast tissue, although they should get close if they're doing shoulders. Someone with more experience might have a different answer though. Just mention it when you get there and see what they say.
Don't be afraid to tell them if its too hard or too soft - nothing worse than cringing through a massage because its just too deep, or similarly wishing she'd put some strength into it.
Don't plan on anything for straight afterwards cos you're going to be very gooey - oil in your hair etc.
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