  In the present work, we study tripartite quantum steering of quantum
correlations arising from two local dichotomic measurements on each side in the
two types of partially device-independent scenarios: $1$-sided
device-independent scenario where one of the parties performs untrusted
measurements while the other two parties perform trusted measurements and
$2$-sided device-independent scenario where one of the parties performs trusted
measurements while the other two parties perform untrusted measurements. We
demonstrate that tripartite steering in the $2$-sided device-independent
scenario is weaker than tripartite steering in the $1$-sided device-independent
scenario by using two families of quantum correlations. That is these two
families of quantum correlations in the $2$-sided device-independent framework
detect tripartite entanglement through tripartite steering for a larger region
than that in the $1$-sided device-independent framework. It is shown that
tripartite steering in the $2$-sided device-independent scenario implies the
presence of genuine tripartite entanglement of $2\times 2 \times 2$ quantum
system, even if the correlation does not exhibit genuine nonlocality or genuine
steering.
