One solution you might consider is "Dropbox" (www.dropbox.com). It provides a free service up to 2 GB of data, and you can pay if you need more space. 2 GB will be large enough for any TntMPD database (which is coincidently limited to 2 GB in size). Dropbox works on Windows, Mac and Linux. You point it to a folder on your computer, and your partner does this also. Dropbox keeps the folders in sync over the internet. You can keep other documents in this folder as well, such as your documents, spreadsheets and presentations used for MPD.
If you're not sure where your TntMPD database file is, read this FAQ article.
I've tested some other similar services, and in my experience they were not safe. If two computers would both open the TntMPD database at the same time, the last person to close TntMPD would overwrite any changed made by the other. I wouldn't feel comfortable using a service like this for my data.
Dropbox is a safe service. If one person has TntMPD open, and the other person tries to open it at the same time, it will block access to the second person. They will have to wait until the first person is done.
If you are offline (not connected to the internet), you can open the TntMPD database file even if someone else has it open. But if the other person changed it on Dropbox since you opened it, Dropbox will created 2 copies of the database. This is good, because it means no changes are lost. But you'd then have to use TntSync to merge any changes.
If you want to maintain the ability to work offline at the same time, you might consider using TntSync. You'd copy the database to a folder not maintained by Dropbox, and you'd synchronize it with the folder maintained by Dropbox. You'd want to limit the times you run TntSync to when you are connected to the Internet.
It also seems that Dropbox is intelligent in how it syncs large files. If you have a very large file, and yet you only change a small part of the large file, Dropbox only has to copy the small part of the file that changed. This is good news if you have a limited connection speed, or if you have a large database.