Can I run Launchpad on my own server

Yes, you can, but keep the following things in mind:

As per https://dev.launchpad.net/LaunchpadLicense, please replace the images and icons with your own.

Also, Launchpad's production configuration information and some configuration-specific admin scripts are not part of the Launchpad code base; you'd have to reinvent those in a way appropriate for your setup.

Finally, keep in mind that Launchpad's code is under very active development, with hundreds of changes released each month. Syncing your private instance with the upstream code base may be risky because your private instance won't necessarily get the same data migration treatment that the main instance gets. Essentially, there's a risk of a private instance becoming an unintentional fork, where its code cannot be safely updated due to the data in the local instance being incompatible with the latest database schema or code assumptions.

Our purpose in making Launchpad free and open software is to enable the community to contribute code review, fixes, and features that will then benefit the entire community by being released back onto Launchpad.net. We're committed to improving our hosted solution, and to supporting the community that is organized around that. Supporting other instances would require significant extra effort (for example, we'd have to generalize admin scripts and data migration scripts), and there would be relatively little benefit to us or to the community we're trying to serve.

So the answer is "Yes, you can run your own instance"... but please be aware of the risks and the lack of support before doing so. We don't recommend it; we'd much rather have you using this Launchpad instance and contributing to its improvement.