The chicken wire is on their property about 1/2 from my 6 foot stone wall.
Disclaimer: The materials provided below are informational and should not be relied upon as legal advice.
Your neighbor can certainly put up a fence on his/her property. The web site of the City of Los Angeles Department of Building & Safety provides: "In most residential areas, fences are limited to 3'-6" in height in the front yard and 6'-0" in the side and rear yards. Depending on lot size and location, the height limit may be increased to 8'-0" for the side and rear yards. The regulations can be found in Section 12 22 C 20 (f) of the LA Zoning Code ." To be sure, you should study Section 12 22 C 20 (f) of the Los Angeles Zoning Code, or contact the Department of Building and Safety for more information.
Construction / Development Lawyer
Most cities will allow 6' high walls, and under some circumstances, higher walls. They also sometimes have restrictions on what kind of wall or fence material may be used. Contact your city to find out what they allow. You could report your neighbor to the city, but if you do, get ready for the relationship with your neighbor to get pretty chilly.