At our hospital, (medium sized level 1 trauma center) Transport, runners, orderlies, can pick up blood from the BB - they need to present patient name/MRN.
For the bedside check, hospital policy states "licensed" individuals - RN, LPN, MD, etc - to verify and sign off the double check. You are correct in that some knowledge of compatibility is needed to sign off.
But pick up from the BB is simply a clerical check - that we are issuing blood for pt X.