hi lynn,
well in my experience anywhere you go as a technician the tech is required to own a decent basic set of hand tools along with a tool chest. Multimeter and the like are generally required to be purchased by the tech but in some cases the company may buy it depending on the company.
As for specialty tools that varies depending on the company.
Most of the lift truck companies i've worked for supply the testers and cables required to work on thier equipment and are generally kept in the shop. In a couple of cases i was issued component testers but was responsible for them and if they were lost i would be required to buy another one. Impact guns etc are usually required to be bought by the tech but in some cases the company might pay to have them repaired if they break.
Really it varies alot when it comes to specialty tools and the company.
A friend of mine works for cocacola and he was required to buy ALL his tools period as far as i know and he has more tools that god would allow lol
But coke pays him very well i'm sure so he can afford it ;o)

right now the company i work for allows us a tool allowance but that is seemingly going downhill, they keep lowering the allowance on us for some reason and restrict where we can use it. Me being a road tech its hard for me to get to those specific places they allow us to use it at so mine does not get used very much. Its easier for me to just go buy what i need where ever it is and if they have the specific brand and type tool i need.