Assertion consumption at SP

Hi,

When Idp issues a SAML assertion and it 
is received by SP. When validates the assertion st SP before granting 
access. Does SP validates the assertion in its own application or it 
makes a call to Idp for assertion validation?

If SP validates assertion locally then how does logout works 
i.e. logging out the session in all the SPs from Idp (Logout Request)?

Thanks.