Chances are it will not heal on its own. As someone said earlier, only a small portion of the mensicus actually gets blood flow which allows it to heal.
I've had three knee surgeries and my advice is to get it done. It's a minor procedure and they will likely have you bearing weight right away.
There are two different types of surgery, repair and removal.
Repair is favorable because it will involve actually stitching the torn piece back together and "scuffing" up the area around the tear to create some bleeding and scarring so it fuses back together as the stitches melt away. The downside of this is that you will not be able to run for 3 months and will not be able to flex your knee past 90 degrees for the first month. Cross training is also out until the flexion restrictions are lifted by your doctor. But at the same time you lose no cartilidge so you don't have to worry about arthritis or anything later in life.
Removal involves cutting out or shaving down the torn area. Depending on yoru tear this could be literally a tiny shave job or removal of 50% or more of the mensicus. This of course increases the risk of arthritis later in life, but having the tear does as well, so you might as well just have the surgery and be pain free for as long as possible. This is a quick heal. You can start rehab right away with no restrictions and some people run in as few as 2 weeks. 4-6 is more likely if you are a slower healer. But you will be able to cross train in the meantime.
As for recovery, your will have to rehab both differently. If you get the repair and rehab well you will feel 100% soon after 3 months and the knee won't feel different once it is healed. If you have it removed you can run in 2-4 weeks but it will take the better part of 6-12 months before the knee starts to feel "normal" again. This is because the newly exposed articular cartilidge needs time to smooth down a bit. The knee will always feel a bit different compared to the other one but in both cases you can return to running with no problem. After 3 surgeries I am still setting PR's and running more mileage than ever.