To I have the following tables:

groupleaders, workgroups, employees

employees,
first name
last name
employeeid

workgroups
groupname
groupid

groupmembership
employeeid
groupid

Thus, a record (3, 4) in the groupmembership table means that the
employee with id 3 is a member of the group with id 4.

Note that employees can be members of more than one group.

So my question is this: is there an easy way with SQL to do a query
that determines the employee who "shares" a group with (i.e. is in a
group with) the most other employees?

Thanks.

I'm not so well in English. So, I may be some misunderstandig of your
requirement.
(It will be better to show sample data and expected result by you.)

This may be an answer, if I understand exactly your requirements.

WITH shared(employeeid, groupid, number_of_shared) AS (
SELECT E.employeeid, M.groupid, COUNT(M.employeeid)
FROM employees E
INNER JOIN
groupmembership M
ON E.employeeid = M.employeeid
AND E.employeeid <> M.employeeid
GROUP BY E.employeeid, M.groupid
)
SELECT S.*
FROM shared S
WHERE (employeeid, groupid, number_of_shared) IN
(SELECT employeeid, groupid, number_of_shared
FROM shared
WHERE number_of_shared =
(SELECT MAX(number_of_shared)
FROM shared
)
)
;