Should I reinforce these joists at the central beam? If so, how?
If you need to remove the 2X2, then you will need to add joist hangers. The joist hangers available have stamped in the steel the nailing instruction for the hanger to perform the way it is designed. Many if not all hangers require "shear nail" in short, toenails as part of the nail sequence in each hanger.

If the 2X2 is not in the way, add cement coated 16D nails (3 1/2") even 12D nails 3 1/4" will work, a total of 5 in each end, 3 on one side, 2 on the other, situated roughly between the 3 on the other side. 

If your span is over 8' (maybe 10') by code you will need bridging, diagonal or solid at the center point of the span. this will greatly reduce the springiness of the floor.