I have the suspicion that some of my saved searches either never update or are missing matching posts. Having timestamps would help narrowing that down.
After checking GitHub, it seems like Type-kun already created issue #2921 for displaying the refresh timestamp. Additionally, he also requested an option to allow a manual refresh for certain levels of users.
Also, if I'm understanding the conversation in that issue correctly, nobody "owns" a saved search on Listbooru. It just has a list of all saved searches, so if multiple users have a saved search for girls_und_panzer, then there will only be one saved search being queried on Listbooru which will be shared with all users subscribed to that issue.