I haven't done this in a while (pre 9/11) but Southwest used to do this in the past. I would be very cautious about doing it at LAX specifically because of the size of the airport and the potential for delay issues due to runway construction, fog... BUT you should definately ask at your city of orgin departure counter. And you will need both boarding passes for sure.
I've been in the travel industry almost 30yrs now and code for Southwest is WN. As a travel agent we had to call them to made reservations when we could use our computer system to book all the other carriers because they Would Not pay computer surcharges to be a part of the system. Also, they didn't have ticketing agreements with other airlines so they Would Not allow us to issue any part of the trip with another portion on a different carrier on their tickets. Lastly, they Would Not transfer bags between themselves and other airlines. They Would Not serve meals. They were known as the Would Not airline to travel agents. They haven't transferred bags...you must have been on a different airline?