Article 38 paragraph 2 of the Vienna Convention seems to apply to such staff members. Per article 1 paragraph c and e:
(c) The “members of the staff of the mission” are the members of the diplomatic staff, of the
administrative and technical staff and of the service staff of the mission;
(e) A “diplomatic agent” is the head of the mission or a member of the diplomatic staff of the
And the text in article 38 (the second paragraph applies here):
Except insofar as additional privileges and immunities may be granted by the receiving State, a
diplomatic agent who is a national of or permanently resident in that State shall enjoy only immunity
from jurisdiction, and inviolability, in respect of official acts performed in the exercise of his functions.
Other members of the staff of the mission and private servants who are nationals of or
permanently resident in the receiving State shall enjoy privileges and immunities only to the extent
admitted by the receiving State. However, the receiving State must exercise its jurisdiction over those
persons in such a manner as not to interfere unduly with the performance of the functions of the mission.
So, under the Vienna Convention, local staff members enjoy limited privileges and immunities.
This is of course a bit vague, I think a better answer may be found in jurisprudence.