I use Dropbox for this. Dropbox contains OpenLP Portable. At home, I add new songs. At church, I wait until Dropbox has synced(!). Then I launch OpenLP Portable and have the updated database. After closing, I also have to wait for Dropbox to sync. Only then can I shut down the PC.
I have no experience with this, but in this case, I would strictly avoid launching the software on both PCs at the same time.
P.S. Of course, any other cloud storage can be used.
Best wishes and God bless!