It's mid-spring right now, and it's the perfect time to start this project as it does take some time before the logs can be used in a wood stove for the winter. We use this method for camping every year, and it saves us the trouble of finding fire wood, or money by not having to buy precut pre-dried fire wood. The benefit is that it's virtually free, provided you value your time at zero.
Step 1: What you will need:
Large container for water
Cotton string or twine
And some spare time
Step 2: Find the perfect spot to setup shop
Step 3: Wet your news print/junk mail
Step 4: Drip dry?
Step 5: Roll it
Step 6: Build it bigger!
Step 7: Rinse...Repeat...forever
Step 8: Drying time
I recommend standing the log on it's end to dry out, rather than laying it on its side.
Wet logs will not burn well, and tend to be smoky, so it's best to be sure to get them nice and dry before you use them. Setting them in a dry location that gets plenty of sun will help dry them faster.
Step 9: Final thoughts
A final note, I advise you be somewhat selective about the kind of paper you use. Some paper like those found in magazines are coated. If this paper is burned it tends to put out a thick black smoke. I don't know what it is that the paper is coated in, but I suspect it's something to do with plastic. This would be bad for the environment, so please recycle this kind of paper.
The best kind of paper to use would be newspapers, and credit card bills(after you've paid them of course)
Here is a shot of the log dried out. An unfortunate side effect is sometimes the paper log will grow a little mold on it while it's drying. This can be seen on the side of the roll. If this kind of thing bothers you, then you should probably stand the roll on a suspended wire mesh, or chicken wire so that it can dry without sitting in a puddle.