Generally, one drops a chain during a down shift. Sometimes it happens only when you shift quickly or under heavy load, but it does happen. There are a few things that can cause a chain to miss the inner chainring: A maladjusted front derailleur or shifter, a very worn or dirty drivetrain, a bent or loose chainring, or a maladjusted chainline (the line which centers your chainrings and cassette parallel to your frame.)
Since your chainrings are also wearing out prematurely, it could be that everything is really dirty or your derailleur is out of whack, but these would be obvious and easy to fix (cleaning and lubing your chain infrequently should get you 2-3,000 miles of life, fastidious cleaning might get you 5k. I have had good luck with wet lubes-specifically ProGold.) Additionally, make sure that any chain that gets installed is specifically for 10 speed systems, otherwise the wider plates may rub against adjacent cogs or rings.
A less obvious cause of premature-wear-and-tear-and-drop would be a bent or loose chainring, but you or your LBS will probably notice this before it totally wears out your rings.
The last issue is one that I think may be worth investigating, and that is a chainline that puts your chainrings too far out. Downshifts will tend to pull the chain to the inside, especially when you are on your inside cog. This could also exacerbate wear from crossing your gears (which, on a compact 2x10 isn't a huge deal.) Any bike that is set up by a pro will spec parts that work (and even a home builder should be in good enough shape. Big problems arise for single and fixed gears, though.) All told, I can't imagine a scenario where a road frame built up with Sram Rival components would have a chainline so far out that it would wear out a pair of rings in 3-4,000 miles. This could be an issue when mixing mountain and road components. Given these instructions, I could also see someone potentially messing up a BB/crank installation.
In any case, a chain keeper-a device which prevents your chain from dropping to the inside-is a small but valuable investment-though in your case not a solution.