Regarding a private organization what would be there options for communication with a probe or rover in deep space such as Mars or farther. Would they be able to use the Deep Space Network? If so what would be the fee? What would their other options be?
One alternative is laser communications. You could have much smaller and more affordable apertures on Earth, e.g. a meter, as well as on the spacecraft, for the same data rates. However you would then need to keep the spacecraft and its laser pointed very accurately to get any data at all. If for some reason there is a failure that results in the spacecraft not being able point its laser to your light bucket on Earth, then it's game over.
The DSN Mission Support Definition lists NASA, other government agencies and international partners as customers (slide 3). The cost structure is depicted on slide 18ff, aperture fee is one of the variable items. Looking at the Mission Service Policies, I would also expect considerable project costs before you exchange the first bit of telemetry with your craft.
At the other end of the spectrum, there are numerous sites with swivel mounted (smaller) dishes, that will work perfectly for deep space, depending on link budget. AMSAT for example is a primary user of the Bochum Radio Observatory (a private initiative!). They were able to receive STEREO telemetry at Bochum, in fact they even helped NASA with the implementation of turbo codes for this mission. AMSATs mars mission P5A is planning to use this dish. I would bet you get some communication time in Bochum if you help to fund P5A.