When they write paid - they usually mean paid in the literal sense of the word. You need to pay for changing towels according to the price list (but the first "batch" is usually given out for free, you can just not change them and not pay). When it comes to collateral, they usually write "deposit". I was also extremely surprised by the paid towels in one of the 5 * hotels ...
I always bring a beach towel with me or immediately buy it in a shop on the spot. Firstly, it is easy to argue for any impudent vacationers that the sunbed is occupied by me and my towel (it differs from the hotel ones) and they were not here, from afar you already find your sunbed with your things and no one steals it.
For example, on the website of one of the hotels they write "The facility at the sea is 123 m long sand-gravel beach, pier and platform. In the pool and on the beach; Sun umbrellas and mattresses are free of charge, towels are provided on deposit basis ."