All of my training is done alone (hikes, bikes, skiing) so I am not totally unfamiliar with being alone, but the JMT is quite a commitment!
You won't be "alone" on the JMT:
When we hiked the trail this summer, one of the things to keep my mind from going numb was to count how many hikers we met on the trial. Heading south, with the main flow, we saw between 20 and 40 hikers each day, not counting people along the trail in camps or at public places like Reds. On Mt. Whitney alone last day we saw 220 people.
We had to go out of our way to find private camp sites, but still spent 7 out of 13 nights with other hikers camped very close to us. August at least is a time of year the JMT becomes a very busy place. Not a bad thing - most people you meet are like-minded and fun to be with. It's the handful of rednecks along the side trails we took that were less than fun to be around (insert mental image here: camo pants, huge campfire blazing pointlessly in the morning, large rambo knives in holsters, and an attitude that we somehow disturbed "their valley" and rangers are jerks and they didn't tell them about the fire rules anyway, so screw you).
October is the time for a JMT hike in solitude. If you can manage the resupply logistics without the regular sites, it's the time to be up there "alone"