I traveled around Italy for 18 days last fall and bathrooms were not an issue. Some were a little harder to find and more interesting than others but not a problem and we did not have to pay anywhere.
We did usually purchase a cup of coffee to use some. We went frequently, if we were in a public place that had restrooms, we used them. I went way more often than I would at home but did not want to have to find one in a hurry.
Some people have suggested McDonald's but we never did that. I don't go in McD's at home so why would I in Rome or Florence? I don't do fast food.
I always carry individually wrapped hand wipes with me and Wal Mart sells a cute roll of travel toilet paper, convenient size.