While OnSong makes beaming songs and sets easy, it does so by loaning songs and expiring sets. This is done to prevent users from inadvertently distributing potentially copyrighted material illegally. Shared songs do not appear in the user's master song list and sets expire after two weeks. Think of this as an automatic self-destruct button to make sure you're being a good citizen.
If you are experiencing this issue, there is no way to undo the effect. You will want to talk to the person who shared the set to see if they can change their preferences to allow this.
First, the individual sharing the songs or set must make certain that they have the legal right to distribute the content. If you or your organization is covered by a copyright license, this shouldn't be a problem. Please consult your legal agreements. OnSong cannot provide legal advice for you on this subject.
If the user is certain that they can do this, have them go into the Utilities menu (gear icon) and opening the Settings Screen. Next, tap on the Menu Settings option on the left and then the Sharing option on the right. Locate the Rights section. Here you can turn off Loan Shared Songs to allow those songs to appear in the recipient's master library. Additionally, the sets can be changed to never expire.