I agree with @resident_heretic that these are the link arms for your stabilizer bars. The bar acts as a torsion spring that tends to keep the car level during cornering.
With the links broken on the passenger side I would expect you to notice a few things:
More body roll during cornering,
A possible loss of traction on the inside wheel in corners, and
Possibly a knocking or rattling sound from the suspension as the disconnected parts bump into each other or into other parts of the suspension.
I found a YouTube Video of the replacement procedure, it looks to be a relatively straightforward process. The parts are sold individually (you can get just the passenger side link arms).
If you elect to do the repair yourself, look carefully at both sides (driver and passenger), for any signs of damage. The stabilizer bar and the links should be symmetrical, any unmatched bends would be a red flag and worth further investigation. The passenger side of the road (at least where I've lived) tends to be rougher, so it is possible that 10 years of driving has resulted in failure on just one side, but I'd still be on the lookout for any clues to the cause of the failure or related damage.