The retired couple has a maid and a gardener, each of whom visit once a week.
Should that visit be visits? Well, yes. Each has a singular meaning, so the verb should be singular as well. But in the process of researching this, I found some interesting stuff about each and when it might not be singular. It's a usage note from American Heritage Dictionary:
The traditional rule holds that the subject of a sentence beginning with each is grammatically singular, and the verb and following pronouns
must be singular accordingly: Each of the apartments has (not have) its (not
their) own private entrance (not entrances). When each follows a plural
subject, however, the verb and subsequent pronouns remain in the plural: The
apartments each have their own private entrances (not has its own private
entrance). But when each follows the verb with we as its subject, the
rule has an exception. One may say either We boys have each our own room or We
boys have each his own room, though the latter form may strike readers as
stilted. The expression each and every is likewise followed by a
singular verb and, at least in formal style, by a singular pronoun: Each and
every driver knows (not know) what his or her (not their) job is to be.
We readers have each her own opinion on this ...