What is the grammar rule for this problem?

We received information that staff of our subsidiaries do not / does not feel well informed.

Thank you for helping out.

Use the plural if you are conceiving of the several members: our staff do not feel well-informed. Use the singular if you are speaking of the body: Microsoft's staff is bigger than Apple's staff.

thanks a lot.
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Staff should be considered a group of something so, unless you're specifying the members, should be "our staff does not" versus "our staff members do not".
Like for team or group:

The staff is in a meeting.

Staff does not reply to individual emails.

The staff members are in disagreement about the findings.

Staff members do not reply to individual emails.

A small comment from Canada.

I seldom encounter the word 'staff'. I usually hear and read 'employees'.

'Staff' makes me think of a retail establishment, like a store or a restaurant.

Students: Are you brave enough to let our tutors analyse your pronunciation?
Good suggestion - replace "staff" or "staff members" with "employee" or "employees" - that should make it more apparent if you're speaking of a single or plural noun.