We consider both local and global theta correspondences for
G
S
p
4
\mathrm {GSp}_4
and
G
S
O
4
,
2
\mathrm {GSO}_{4,2}
. Because of the accidental isomorphism
P
G
S
O
4
,
2
≃
P
G
U
2
,
2
\mathrm {PGSO}_{4,2} \simeq \mathrm {PGU}_{2,2}
, these correspondences give rise to those between
G
S
p
4
\mathrm {GSp}_4
and
G
U
2
,
2
\mathrm {GU}_{2,2}
for representations with trivial central characters. In the global case, using this relation, we characterize representations with trivial central character, which have Shalika period on
G
U
(
2
,
2
)
\mathrm {GU}(2,2)
by theta correspondences. Moreover, in the local case, we consider a similar relationship for irreducible admissible representations without an assumption on the central character.