In New Zealand they use the term "bring a plate" in written or spoken dinner invitations. It means a dish to pass, or something to share. When I first heard it I had a micro-second pause, but understood what they meant. But occasionally some people from overseas bring a plate. Just a plate, usually with a very quizzical look.
I understand in America it goes without saying that the guests will bring something, unless told specifically not to. So "bring a plate" seems to be a whole cultural shift, because it implies that if guests are not told it, they should bring nothing.
I hate to confess it in case it reveals I'm a Philistine, but no one ever taught me that. I see that in some American sub-cultures a guest may bring a bottle of wine or some other non-essential when invited to dinner—see it in the movies, I mean, not in any family of my personal acquaintance—but it wouldn't occur to me to bring a dish of food unless it was specified. In fact, given how many people I've met who are far too eager to be offended, I predict that some would take it as an implied insult ("what, you think the food I provide my guests won't be good enough?"). At best, whatever I bring might clash with the selection planned by the host. If Americans generally fall on hard times, this could easily change, I think.
In my own family (where we understand each other very well and, God be thanked, there seems little need to walk on eggshells), I would feel free to contribute something. But even there I wouldn't assume, but ask ahead of time, confident that my siblings would feel free to say "yeah, if you feel like bringing some salad makings, that'd be great".
I don't know what part of America you're in, RobertB, but expecting guests to bring a dish? I've never heard of that before. For a picnic or standard potluck, yes. But not for every invitation. And even if guests show up with a dish, according to etiquette rules, the host need not use the dish at that particular meal.
As for family meals or meals with close friends, we do much as Bob Bridges does with his family--ask if anything is needed. But I would never assume to bring a dish unasked.
I'll ditto Bob and Jackie. Often, what we call a "hostess gift" is brought to a dinner as an appreciation but it is never food for the dinner unless that is prearranged.