When given steroid injection for trigger thumb, will steroid cause rise in my bp or hr? i'm hoping the effects stay mostly local and will not be systemic?

Localized steroid injections into joint spaces rarely contain enough steroid to disperse later in the circulation and have any systemic effect. The joint space itself is isolated from direct contact with the circulation making such targeted treatments more effective.

Agree with above. In addition, steroid injections for tendinitis (trigger finger/thumb, ECU, and Dequervain's) rarely cause systemic side effects. Locally, hyperpigmentation + atrophy can occur.