Here is a site which discusses DIY sharpening:
http://geography.hunter.cuny.edu/~mc...Sharpening.htm

The site also mentions a simple method for testing the
sharpness of the mower blades.

Below, I've cut-and-pasted some comments regarding why
you might not want to try it yourself. Of course, this is from
a company which sells the service that they recommend
you farm out.

Good luck,
Gideon

Welcome to our Do-It-Yourself Web Site and my Sharpening
forum.
Not really an art but it is a skill. Must be professionally sharpened.
Best left to the pros to do.
One of many difficult tools to sharpen. It takes a specially designed
and built machine to sharpen reel mower blades. So much so in fact
that the machine does only this sharpening procedure.
Many sharpening shops do not have the machine. Most lawn mower
repair shops should have such a machine but even this is not
guaranteed. The machine is costly and takes up lots of space.
On most of the commerical sharpening machines, the reel has
to be removed from power reel mowers. Hand push mowers the
mower is mounted upside down. A power grinder wheel is used
to sharpen the blades.
The bed knife must also be removed to correctly sharpen it. Once
all blades and knives are sharpened and reinstalled, the bed knive
has to be adjusted to the blade so the reel cuts against the
bed knife like a scissors.
The process is explained above so you and other readers of the
question can understand the process and appreciated the labor
required and costs involved.
So everyone can realize that many claims can be made by other
types of sharpening methods and machines but none can compare
to commerical power sharpening equipment designed specifically
for this purpose.
In my professional opinion, no other method can produce the
wanted and needed results. Reel blades and bed knives cannot
be accurately nor properly sharpening by hand. It's worth every
penny to have a reel mower professionally sharpened.