Two Different Sources Taking Transaction Log Backups
If a disaster occurred and I needed to recover the database from backup, would I be missing data?


As long as all of the backups are in tact, no.

The transaction log chain is not broken, and point-in-time recovery is possible.

It's just that the backups that constitute a complete transaction log chain are not all in the same location.

Having said that, I would recommend consolidating the systems so that only one of them is responsible for taking backups. While it sounds like your situation was set up that way unintentionally, if you want to create a redundant copy of a backup at an alternate location, there are many options to do so, but that's beyond the scope of the question here.


  If I were to do a restore from either backup (unless it's using only the full backup), then it would not include all of the data.


Correct, you would not be able to restore from the TDP backups alone, or the SQL Agent backups alone.

After restoring the latest full backup, you would need to restore all of the log backups between that point-in-time and the current point-in-time. This will involve combining the set of backups from both locations, and restoring them in the correct sequence as each log backup will contain a distinct portion of the backed up transaction log.