I have a sports massage regularly, and always strip down to my knickers. She works on the muscles at the top of my glutes, and my periformis, and needs to have a good go at them!
I have seen pics of sports massages where the person is wearing their shorts. Which would be fine for lower legs, and possibly most of the quad/hamstring, but for anything lower back/glute related I think they'd get in the way. A sports massage is DEEP - I'd find that odd to be done through layers of clothes - how can the therapist get a nice smooth action into a focussed spot when there's layers of fabric in the way?
Which part(s) of your body are you hoping for them to work on? A sports massage isn't like a Swedish massage or full-body massage at a beauty salon. They focus on specific areas and work them hard. I go for an hour, and in that time I usually get my calves, glutes and shoulders done (or maybe calves, back and IT band, or another combination of niggly areas!). Most of it is done with me face down, so I don't have to turn over mid-way, but if I do, she holds the towel up such a way that I can turn.
If it's general stiffness/aching that you're suffering with, you might be better with a normal massage, rather than a sports one. They can still massage deep, but a sports massage is a much more focused/specific treatment.
Having said all that - your body is not repellant to a therapist. Far from it. There is NOTHING they won't have seen before, and they want you to be at ease/comfortable.
Can you get a recommendation from somewhere for a good place to go? Knowing something about the therapist might put you at ease.