You can save searches in Bugzilla, and even share them with a group.
To save a search
While in Bugzilla: Just trigger a search. Then type a name in the "as" textbox at the bottom of the search > Click "Remember search"
You will see it now at your footer in the Saved Searches row as a quick link.
Share your search
Assuming you have already saved some searches:
Login to your Bugzilla instance > Click Preferences (middle top) > Saved Searches > Edit the "Share with a Group" value for each search.
There you can select the Groups you like to share you search with. And also manage how to use Searches shared by others.
Your shared searches will appear at the Saved Searches footer of your fellow group members, just like the ones you saved for yourself.
Still can't share any searches
If you (or some of your users) cannot share their searches, you might need to add that group to the "querysharegroup".(you need to be an admin for this)
Login to Bugzilla > Click Parameters (bottom left) > Group Security > Edit querysharegroup as desired.