The theatre I work at, most, does not have a secure space for transient (SM/ASM/designers) personel to store belonings or paperwork. I've asked for something, a locking filing cabinet, a drawer backstage, anything, several times, but as of yet, my requests have not been honored. (Also, the "valuables cabinet" is only 4" deep and a joke.)
There is a limited number of people who have keys to the booth, so a lot of the time, I'll put my stuff there, if I absolutely have to leave it. However, it's such a pain to transfer books, kit, laptop, laptop bag, sound computer, etc. up to the booth, most of the time I end up staying in the theatre over breaks, to guard stuff, instead of going out. It sort of galls me, but, there it is. I've resorted, more than once, to shoving my valuables under a seat in an out of the way section of the house, and praying.
Once we're into tech, the prompt book stays at the theatre. It lives on the tech table so designers can refer to it when I'm not present, and then lives in the booth the duration of the run. The booth is pretty secure, but it's such a mess and so overcrowded with other stuff that storing anything much more than the prompt script isn't an option.
I do keep all paperwork (contact sheets, reports, health forms, etc.) in a separate book than the script, which stays with me. I've never really felt that there was a security issue with this arrangement.