Can you give some examples of where you find old free laptops?
Since you asked, here's my rambling stream-of-consciousness reply:
The first rule is simple: ask around! Mention it to people in your network, mention it to your IT guy at work, mention it to friends, mention it on CraigsList or Kijiji.
Be direct and up-front: "I am learning how to recycle and repair old laptops. I install a free operating system called Linux. I am not reselling these in any way. When I am done, they will be provided to people who need them or properly recycled."
If people are worried about privacy (and they should be!) offer to remove the hard drive on the spot and drill holes in it with a cordless drill.
IT departments often discard flats of laptops when the OS goes obsolete. It's cheaper to buy new than to upgrade them. This certainly happened with Windows XP -- I scored half a dozen IBM ThinkPads (T60s and T61's) with good chargers, moments before they went into the recycling bin. The rougher ones I gave to hobbyists or scrounged for parts. But I still have one of each, and with memory maxed out and solid state hard drives, they scream along perfectly well. They will probably still be ticking along after the next few generations of consumer-grade disposables have conked out. I think they have the best keyboards every made.
Hoard old chargers from your dead laptops. Thrift shops are great sources for laptop chargers too. Dell chargers in particular seem to be everywhere -- corporate leftovers I presume.
Check your area for not-for-profit recycling societies where you can volunteer your time, learn new techniques for refurbishing old laptops, and be rewarded with a refurbed laptop as a reward for your volunteer efforts. This is brilliant.
The other source is rather controversial, and might offend some here: I do not hesitate to discreetly remove discarded laptops from recycling bins. In my mind, these have been discarded in the same way as trash in a dumpster, and I have no ethical qualms whatsoever about reducing our mountain of e-waste by a few pounds.
Recycling centres are making this increasingly difficult, but I have found out it's not because the hardware is worth anything (it's not). It's because of liability concerns. If staff notice it, they are obligated to put the kibosh on it. I respect that, and never put them on the spot.
If I do snag a laptop in this way, a lot of careful cleanup and testing has to be done. Often, I'm more interested in the RAM for other machines, or the wifi card. If I have a compatible power supply, I'll run a low level drive wiper called DBAN from CD/DVD and then a self-running memory test program (MemTest+). This gives me some idea if the device is at all salvageable, even for parts. Safety is key here, as an unknown machine could have electrical hazards, illegal material, or all manner of horrors. It's not advisable for anyone without technical skills and a strong stomach. Since I have a suite of Linux hard drives already, I will often destroy the hard drive and start with the MemTest. A mildly soapy rag, maybe an alcohol wipe, blasting out the crud with compressed air, and a keyboard/display test will hopefully bring it to life. It's often helpful to reset the RAM modules in their sockets for reliability.
I would never give someone a scrounged laptop until I had used it as my "daily surfer" for quite a while -- watching for heating and problems closely. But by connecting all these dots, I have provided clean, functional laptops to quite a few people who need them but are struggling. It's pretty hard to function these days without some sort of access to the Internet.
That's my method. It works for me. Whether it works for others, practically or ethically, I cannot presume to say.
Side note: Opera for Linux now has a free VPN built in -- and it works surprisingly well. I don't know whether I would trust it for banking on an otherwise open, public network, but it's good enough for puttering around in coffee shops.