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Currently Visio is not offered to Mac users. So there are two options if you want to run Visio on a Mac. The first is downloading crossover software and then download visio which can be costly and obnoxious in my opinion or if you still want to work on and download visio files, Lucidchart offers a pretty good offer for free to view and edit visio files. It also doesn't require downloads which makes it nice and easy to work with.
If your company is open to cloud apps, I'd encourage you to look at Lucidchart. Along with being like a Visio for the Mac, it's cross platform so it's also like Visio for iPad. It can import/export Visio files which makes things a bit simpler as well. Have not compared it directly to Omnigraffle though to tell you the key differences.
Omnigraffle is your best bet, for what you're looking. The standard version is $99. The Pro version, with Visio support, is $199, though - still cheaper than Visio's $249.
In the Pro version, in addition to opening Visio's VSD file, Omnigraffle will also open Visio stencil (VST) and Visio template (VST) files.
(I do not work for Omni. I do use a Mac, and Parallels whenever the job requires otherwise.)