When an OnSong user beams a set, those songs are marked as loaned by default. This is to prevent you from unknowingly distributing potentially copyrighted material without permission. To keep you on the "up and up," loaned songs do not get permanently copied into the recipient's library. Instead, they appear within the set that was sent to you. Additionally, these sets are flagged to expire after two weeks by default. This is likely why those songs are being removed or unavailable in the recipient's library.
If you have the right to distribute the songs you are beaming, you can go into the Utilities Menu by tapping on the gear icon and choose Settings. Then go into the Menu Settings section and choose Sharing on the right. There you can disable the Loan Shared Songs option as well as change the expiration of sets if desired.
If you've received sets or songs from someone, you will need to show them how to change this preference and send the songs again.