You state part of the answer in your question. The people in our social circles give us a better picture of who we are and what we're capable of.
For strengths, one easy way is to determine what hobbies or jobs you enjoy and do most. It's unlikely to be awesome at something you hate doing. The world, just doesn't work that way. Which I suppose, is a good way at finding some weaknesses.
Ultimately though, it depends on what you need this knowledge for. A career path? Entrepreneur ideas? Hobbies? Making more friends? Finding dates? Personal growth, always needs some underlying reason. If there's no reason, then the mind is just getting wrapped up in a trivial matter.
I think what people need to do, is make a personal report card. Listing all your achievements and past accomplishments (minor and big, whatever makes you feel good inside). This, should paint a clear picture of what your capable of.
Writing present goals for the future, will also remind us what strengths we need to accomplish them. We won't know what our weaknesses are, without testing. Which is why, goal setting and this report card is so important.
If someone wants to know weaknesses and strengths in socializing without a third party, keep a journal. Take notes of the effect you have on different people. People you know and strangers you get in conversation with.
At least, those are some of my ideas for understanding ourselves better then before.