The original stock markets evolved from coffee houses where investors would meet and swap paper over coffee. Gradually, as volumes grew, the participants at the coffee tables became agents for others. Eventually, the coffee houses were replaced with full scale trading floors which evolved into the sophisticated web-based trading systems we use today. Well, you can still do it the old way, and I sometimes do. It's a slow, somewhat painful process which instills (in me anyway) a sense of nostalgia and history. Trading shares directly over the table with family and friends can save you a little money (no transaction costs!) and it can be fun.
For several years I've held stock certificates directly in firms that have a good DRIP or share purchase plan. I don't hold many investments like this--I strongly recommend to you the excellent indexed mutual funds and exchange traded funds (ETF's) that are available through your discount web broker, they're much better than holding individual securities. Still, taking advantage of share purchase programs in well diversified REITs and other trusts can save trading costs--in some cases you can simply mail a cheque in every month and receive shares at a slight discount to the prevailing market rate, and with no trading fees either. Usually I buy these shares the traditional way: I execute an electronic trade through my broker for a few shares, then pay a small fee to have the certificate sent to me. Once you've got your initial few shares you can participate in the share purchase program, if there is one.
In a few cases, though, I've done it the old way: I met up with some friends for coffee, and we traded paper over the table. I handed over $160 cash, and recieved a certificate for 10 shares from my friend, filled out in my name, which I mailed in. It wasn't quite as simple as that, though--you have to get the signature on the back of the certificate duly witnessed by an officer at a chartered bank. It's also important to note that I exchanged these shares with a close friend--not the "general public". To offer securities to the general public you have to be a registered broker. Still, within the limits of friends and family you can still get together and trade stocks over coffee.
In the past when I've done these transfer I've mailed these forms in to the transfer agent, but recently I transferred a certificate over to my wife and decided to save $0.50 in postage by walking into the transfer agent's office myself. It turned out to be an interesting experience: CIBC Mellon was the transfer agent in this case, and their office is in some dungeon under Commerce Court. You take the elevator down to the sub-basement. The transfer agents sit behind a big glass security wall, and you pass your documents to them through a little window, which they carefully inspect, and then duly update an entry in their books with a fountain pen (ok, ok, they type it into their computer). Ordinarily they receive packages couriered from brokers, but they'll serve you too if you walk in with the proper paperwork.
The back of your share certificate has a form for executing transfers--fill it out with the name of the person you're transferring to, address, dates, and the like. Oddly, you cannot use your ordinary signature to sign the back: You'll have to write your name out clearly, so that the transfer agent can read it and verify that it's exactly the same as the name on the front of the certificate. You'll need to have that signature guaranteed by an officer at a chartered bank who will visually check your ID, and stamp the share certificate with a "medallion stamp". You'll also need a "letter of direction" but don't assume you can write that yourself--the transfer agent has a special form letter they'll want you to use, and if you try and write your own letter they'll probably reject it. Once you've got all the paper work properly filled out, though, it's pretty easy.
The most interesting part of the process is how the transfer agent checks the signature guarantee: They've actually got a book with the signatures of every bank officer in Canada, and they flipped through it to find my branch manager, and visually checked his signature. Since I'd done my paper work properly, the process took under five minutes and all was good: Two or three days later my wife received a freshly printed share certificate in the mail.
There's no major point to this story--I do not recommend you give up on indexed mutual funds and ETFs, nor do I recommend you give up on your fast, efficient, discount web broker. The process of getting an appointment with your branch manager, physically exchanging the paper, and having it registered is slow and somewhat painful. Also, even in this case the majority of my wife's holdings will be held in a computerized virtual account by the transfer agent--only these initial ten shares will be held on paper, shares purchased through the SPP and the DRIP programs will all be held on account the modern way.
Still, the process exists, is sometimes useful, and to me anyway it's an interesting glimpse of how things used to be done.