You can! qbittorrent can manage both private and public torrents at the same time. You can also cross seed with it, but it's more involved than checking a box or clicking a button.

The main issue with private trackers is getting in. Most of them need an invite from a user that is already inside, and you can't ask for invites or ask them for an account.

Some trackers have open signup days, and some others require you to go through an interview process to get in. After you're accepted to those, you can gather reputation and user level, and eventually, access to invite forums. In there, you can be "recruited" to more exclusive trackers.

It's a really interesting concept, and despite its closedness (or thanks to it), it really is night and day with public trackers. You can find a lot more stuff, torrents rarely die, you can request new stuff, you get fast download speeds, and everything is way better tagged and organized than on public trackers.

If you're curious about music trackers, REDACTED (RED) and Orpheus (OPS) both usually have interviews open. You can read more about the process and the interview material here for RED and here for OPS.

Additionally, I've heard MyAnonaMouse (MAM) has a really friendly community for starters. It's mainly a book tracker, but it also has some music. It regularly opens registrations, info here.

There are also some general content trackers around that sometimes have open registrations, but the registration windows are short and random. You can subscribe to !trackers@lemmy.dbzer0.com , where people post open registration windows for trackers.

I can't imagine any reason anyone would care what torrents are being seeded by a private tracker user outside of that private tracker