Having publicly running web application always brings challenges in terms of security and in generally in handling untrusted data. Security wise Weblate has been always quite good (mostly thanks to using Django which comes with built in protection against many vulnerabilities), but there were always things to improve in input validation or possible information leaks.
When Weblate has joined HackerOne (see our first month experience with it), I was hoping to get some security driven core review, but apparently most people there are focused on black box testing. I can certainly understand that - it's easier to conduct and you need much less knowledge of the tested website to perform this.
One big area where reports against Weblate came in was authentication. Originally we were mostly fully relying on default authentication pipeline coming with Python Social Auth, but that showed some possible security implications and we ended up with having heavily customized authentication pipeline to avoid several risks. Some patches were submitted back, some issues reported, but still we've diverged quite a lot in this area.
Second area where scanning was apparently performed, but almost none reports came, was input validation. Thanks to excellent XSS protection in Django nothing was really found. On the other side this has triggered several internal server errors on our side. At this point I was really happy to have Rollbar configured to track all errors happening in the production. Thanks to having all such errors properly recorded and grouped it was really easy to go through them and fix them in our codebase.
Most of the related fixes have landed in Weblate 2.14 and 2.15, but obviously this is ongoing effort to make Weblate better with every release.