Hey I live in LA so I am on a first name basis with most of the local "dumpster divers". It is a problem of course but very difficult to discourage under most circumstances. And actually there are very good reasons why we might encourage such salvage and recycling so look at this as less of a problem and more of an opportunity to do some good.
here are some tips for you:
1. Generally speaking digging into garbage placed in the public right of way is not considered a "search" of your private property requiring a warrant or even permission. (Although local ordinances may address non-law enforcement or sanitation worker access.) Therefore, the police or anyone else may sift through it at any time so naturally if there is something they shouldn't find don't dispose of it this way.
2. Be a good eco-citizen. If you don't recycle make it easy for others to do so. Separate your aluminum cans and bottles into a separate container and place it where the trash diggers can get to it. Incidently, these articles are primarilly what they are looking for. If you don't want them making a big mess and taking forever to dig through everything by doing this you will actually speed up the process. If it become regularly understood that you will place all such things in a separate box or bag for them, when they see that it is empty they may leave right away. And again it is better to encourage recycling than to discourage it.
3. Anything that is large and merely used, old, slightly broken or otherwise in working condition, either don't dump it--take it to the salvation army or some other place where they will let you donate it (good tax deduction) or if you are too lazy to do that place it on top or beside the garbage. That way again the divers can just take it and go without making a mess.
4. Finally get a small paper shredder (they are very cheap now) and make sure you shred all credit card statements, financial records, bills, etc. etc.--anything that may contain your name, DOB, SSN, address or other private information.
5. Leave some WT literature by the dumpster and a note offering a "Homeless" Bible Study (only don't use the word homeless since not all divers are). Offer to make a "return visit" the following week when they come by again trash digging. Your study with them will ultimately lead to them becoming respectible and hopefully not having to dig through your trash any more.
--either that or you will soon notice that they begin skipping you on their route.