I have made another alternative account for my website, and I use it in a different browser. When I enable all notifications via settings, and enable web notifications via the admin panel, nothing gets notified. Even if I accept a friend request, nothing is notified in the notifications for either party.

[Edit: Fixed by installing Cron job]

Hmm, do you know if you set up the system scheduler (Cron job)?

See

Composr Tutorial: Basic configuration and getting started

Open Source CMS with advanced social, interactive and dynamic features. Fully flexible, themeable and extendible: suitable for building powerful websites.

View

(Scheduling) for more information.

Some notifications rely on the scheduler to be running to work. I'm not 100% sure this is the issue but it's worth starting there.