I'm left handed. Always have been. Always will be. I sometimes have lunch at a local Chinese restaurant. And, in the course of said lunch, I often get my glass of ice tea or whatever refilled. Without exception, they take my glass which is on the left side of my plate and return it to the right side of my plate--it never seeming to occur to them that I must want it on the left side or it would not be on the left side. I actually half expect them to come up and move it over whilst it's still full.
While on the subject, I can do must things right handed except write or throw something. I can easily eat, drive, open a door and even shoot a firearm right handed (though accuracy suffers greatly in the shooting category). The notion that often pops up in mystery stories that if you unexpectedly toss something to a leftie he'll only be able to catch it with the left hand, thus proving he's a leftie, is preposterous. It's not catching, it's throwing where the difference comes into play.
Alas, it's not just the Chinese place that insists on placing my refilled beverage on the right side. Try it sometime. Putting you drinking glass or coffee cup on the left side of your plate will drive many servers crazy.