I've been living, working, and, most importantly, hiking in Yellowstone all summer. Almost all of my trips have been done solo -- it's surprisingly hard to find people interested in anything more than a 5-mile day hike.
I have seen bears. I saw a grizzly from very far away around the Tower/Roosevelt area. Hiking to the summit of Mount Holmes, I saw a grizzly in hot pursuit of a group of three female elk. The group ran about 100 yards to my right, cut in front of me through the trees, and headed 30 yards down my left to a meadow where the bear finally caught the last elk in the line and took it down. This was truly frightening.
I say this not to scare you, but to emphasize two facts. 1. All of the rangers and long-time Yellowstone workers to whom I've told that story were absolutely amazed because it is VERY RARE. 2. Encounters with bears are a risk in Yellowstone.
Statistically speaking, hiking solo increases the risk of an encounter with a bear, and, obviously, should any kind of injury occur, you have no one to help you get out. However, if you are alert, carry bear spray, make sure to make noise, keep track of your position in the wind, are generally a smart and well-versed backpacker, make sure to carry low odor foods in airtight, odor proof bags that you always hang properly at night, and have a well thought out plan for encounters with and possible injuries from bears, your risks are very low. You have a much better chance of being killed in a car accident than you do of having any kind of negative encounter with a bear.
Ultimately, you have to decide if the benefits of solo backpacking in the area are worth the slightly increased risk. From my own 3 months of experience in the park, I can say that for me, yes, solo backpacking in Yellowstone has been fine. If you're curious about what areas tend to have more bear activity, look for the Bear Management Areas on your maps, consult park rangers, and have a general understanding of what types of habitats bears tend to prefer. I have not hiked much in the Bechler region, so I can't comment on it specifically. I have been in the Beartooths and have not personally encountered any sign of bears.