Each institution will likely have different guidelines for using their spaces. You may wish to check with them in advance to make sure you are not surprised about what you can and cannot bring into the Reading Room.
In general, you can likely expect to not be able to bring in much of your own possessions aside from a laptop, cell phone, and/or digital camera. At Princeton, all other personal belongings can be locked in a secure locker outside of the Special Collections suite. You will also likely be given care and handling instructions in advance of using the collection materials. These instructions may change depending on the format of the item or its fragility.
Be on the lookout for any announcements on official websites about changes in access
When in doubt, reach out to the library. Princeton University Library's Ask Us! form is one way for our own visiting researchers to contact us. See if there is a contact e-mail or form to send questions to at your archives and libraries of interest.
Access the finding aid ahead of your visit and learn how to request material:
Visit Us!
Showing ID - How many forms? What type? Original or copy?
Availability of Materials - appointment required
Are you finding the right info for researchers vs. visitors to the building
Do you need permission to access certain collections?