Does a pain specialist give steroidal injections or will a neurologist have to do that?

Trained physician. These injections are performed under fluoroscopic guidance (x-ray) by a highly trained and experienced physician. Pain specialist, neurologist, anesthesiologist is first on the list.
Pain specialist. A pain specialist can, sometimes orthopedic surgeons do depending on part of body.
Depends. Usually a pain management doctor gives steroid injections, but depends on the qualifications of a neurologist.