I want to know if auto insurance companies all have access to the same
claims history, like credit companies have access to the same credit
reports. I'm deciding whether to make a claim or not and want to know
if my premiums will rise relatively equally across a range of
companies or if they will rise more with the company that I'm
currently with and less, if at all, with other companies.

All the companies will get the information.  If the claim was an
accident that was your fault then you'll definately want to consider
the costs/benefits before claiming it.  But if it is a matter of luck
that your car was damaged in some way then it will not affect your
insurance rates much.

Be sure to consider your deductible so you know how much money you're
entitled to receive so you can compare it to how much you'd expect
your insurance premium to go up.  I had to replace my windshield once
and with my $250 deductible, I recieved a nifty $19 from insurance for
my $269 windshield.  I know for certain the paperwork and potentially
slightly higher premiums are not worth the $19.