  In the present paper we prove that every 2-local inner derivation on the
Jordan ring of self-adjoint matrices over a commutative involutive ring is a
derivation. We also apply our technique to various Jordan algebras of infinite
dimensional self-adjoint matrix-valued maps on a set and prove that every
2-local spatial derivation on such algebras is a spatial derivation. It is also
proved that every local spatial derivation on the same Jordan algebras is a
derivation.
