How can I tell if the wood turning chisels are sharp?
As with any edged tool you can simply feel the edge (carefully) to check if it's sharp. Almost all traditional woodworking books show a chisel or plane iron's edge being testing with the thumb. However, it can be difficult to judge just how sharp something is by feel — sure it's sharp, but how sharp? — so more empirical tests can be a better idea. So directly testing the edge for cutting performance would be my recommendation.
One empirical test that is often used today is to see whether a tool will easily shave hair from your forearm, which is a particularly good test of a high degree of sharpness but it is not ideal as you can easily run out of arm hair if you are testing many edges over a short timeframe. And there is obviously the problem that not everyone would find bare patches on their arm acceptable in the first place :-)
Probably a better test is to directly use the tool on wood. Paring the end grain of a softwood such as pine is a very good acid test of sharpness. You do this with a softer softwood and not a hardwood because the soft parts of something like pine will easily bend over (showing as tearing of the fibres) instead of being sliced cleanly. A good edge will slice pine end grain and leave a uniform surface behind, if it's very very sharp indeed the surface will be nearly glossy.
In case you are unaware, woodworking chisels usually need to be sharpened or honed frequently in use, as often as a few times an hour. As well as that, few are supplied sharpened and ready to use so they need honing before their very first use too.
Two related threads that might be of further help:
What should one consider when purchasing an intro turning chisel set?
What are these blemishes called and how do I avoid them?