You must use listed connectors to connect the ground wires together (i.e., wire nuts). Twisting alone is not sufficient. You must connect the ground wire to the device (except for certain devices specifically designed to ground themselves to the mount). You must use some approved method of grounding the metal box. Using the device mounting screws is NOT one these.

Thanks, This is going to be more difficult than I had hoped. The house was built in 1949 and I suspect that the second or third owner put in, or re-cabled, the circuit that I am describing. That was probably in the 1960s and he was an electrician. Was simple twisting of the grounds permissible at any time?

Obviously, I'd like to improve the situation while installing the grounded receptacles, I don't, however, want to separate the metal boxes from the studs and install new ones. I have to check more carefully to be sure about the box attachment, but I don't think that they are a type that clip on to the cement board/plaster.

Is there a clip that I can use to properly ground the old metal box? I recall mention of them. There may be room in the box for a wire nut, but these are small boxes. :-(

The circuit powers some outlets and lights in adjacent rooms, but I've removed an outdoor outlet and two indoor outlets from it. (They are now part of a new circuit.) The two indoor sockets had a single, green ground wire and a two-conductor cable leading to a junction box in the crawlspace. The ground wires were twisted together there and corroded badly. At least the twisted grounds inside the house are clean-looking.