I'm not intimately familiar with the 350s, but I was able to shave my 360 down to 284 pounds (dry). I guess it's closer to 300 with all the fluids.
Here's a list of the things I removed or replaced. It's only from memory so there may be some extras. Check my build thread for details:
*Rear passenger pegs and frame mounts removed
*Stock airbox and intake system replaced with velocity stacks
*Battery and air box mounts removed
*Center stand mounts removed
*Chopped the rear and replaced with steel hoop
*Front fender removed (running just the brace)
*Rear fender removed and replaced with a thing brass sheet
*Stock seat and tail removed (along with the mounting tabs) and replaced with fiberglass
*Removed tach and horn
*Stock dash stripped off
*Wiring harness stripped out and replaced with custom wiring
*Replaced wind sail turn signals with small LED versions
*Stock 2:1 exhaust system replaced with lighter 2:2 version (Don't ask me, I have no idea why two pipes are lighter than a 2-into-1 system, but stock one was fuggin' heavy!)
*Stock bars replaced with steel clubmans (I'll go to clipons, eventually)
There was also a lot of assorted minor stuff like the move to forged pistons (11g saved, each), shaving the casting marks from the rocker arms (5g saved, each), and stonger valve springs with aluminum retainers (21g saved, each). Longer term plans to remove more weight include the move to aluminum wheels, venting the drums, drilling out the front sprocket cover, switch to aluminum rearsets (to grind more of the frame away), suspension upgrades, and more internal engine work (cutting the gears, balancing the crank, etc).
Here's a pile of stuff that came off my bike with only a few pieces going back on:
And here's how the bike currently sits:
Not the prettiest thing in the world, but very functional. That's why I choose the name SOS for my project, "Substance Over Style".