Certain websites, like mostlymusic and jmusic.me allow you to download current and past Jewish music singles for free. Can we assume that they have some sort of arrangement and that it is not stealing to do this?

ב"ה Shalom, Your integrity is praiseworthy in looking into the legality of websites in regard to the use of music. However, I as a Rabbi cannot help you much. Because I'm not familiar with these websites and neither would it be correct for me to assume anything in regard to any of these free less known music websites. If one of the readers could tell us how to verify these things , this person would be "mezaceh the rabbim" cause merits to the masses, by preventing them from using music which shouldn't be used and on the positive side it would give support to those who share only music obtained legally. All the best