You can send a query which consists of a logline and short background of yourself. Productions don't want scripts sent because it is a major legal headache. Yes, in fact, there are lots of scripts that share the same story idea. They have no wish to be accused of stealing an idea which they have already legitimately acquired. Get involved on sets. Often personal involvement builds connections that can lead to getting scripts read. If you have an agent (as a writer or actor), they can be very helpful. One good way of getting 'solicited' is to write and shoot a couple shorts with your friends. You can then include links to youtube, vimeo, etc. in your query letter. If you have credits to your name, it is easier to get a producer's attention. However, don't expect a storm of requests.
Another way to do it is to list a short on Inktip (can be done for free if under 40 pages). It will sometimes get you noticed. Sometimes the big script you want to sell is not the one you should lead with. If you can build a bit of reputation, you'll be able to get more requests. Good luck.