… a couple of landline phones? What would be easiest—throwing them into the dumpster—doesn’t seem right, and there are friends’ and family members’ phone numbers programmed on them. I know there are plenty of places for used cell phones, but landlines?
I’m at a loss about how to dispose of them. 
If they were the old phone company-supplied dial phones, I’d take them off your hands. But they’re not. Probably the easiest thing to do would be to take them to the Salvation Army or Goodwill. Homeless shelters might use them too. Shelters will try to get people housing, and those people might eventually need a phone. (I donated a set of cookware, and was told that they would be given to someone the shelter finds housing for.)
See if your municipal trash/recycling organization has a website, and check there. My county has a nice website that you can just put in phone, and it gives you a list of recyclers or reusers.
Before you get rid of it, erase the numbers, or reprogram them to a nonsense number that won’t go anywhere.
I bought one within the past two years from Goodwill. Seems like a good place for that and VCRs these days.
Yes, that’s part of the problem – I have nothing to hook them up to anymore so I can do that. Sorry I didn’t mention that in the OP.
Why? I mean, I fully understand not leaving an electronic trail, and I’ve power-wiped the hard drive on any computer I’ve donated…but a landline phone? Is the ability to speed-dial “Mom” and “Vinnie’s Pizza” going to get an identity thief somewhere?
No – that’s true, but might some ornery person not get a kick out of bugging the hell out of Mom and Vinnie?
It’s been awhile since I played with a landline smart-ish phone, but I think the speed dial memory is maintained by batteries in the phone, or a wall-wart independently from the Telehone Co’s signal.
So I bet you can reprogram all the numbers to be zero whether or not you have a dial tone to plug in to.
Heck, if you find the phone does have 2 AA or whatevers in it, just pull those & take the phone to the Goodwill without batteries. Whatever numbers may have been in there will be gone in a few hours tops, if not at the very instant the batteries are removed.
Also, seconding the idea that worrying about numbers left on a speed dial is just silly.