Kruger National Park is huge! Additionally there are private game reserves that are adjacent to the park. The animals roam freely; there are no fences around the park or between the private reserves and the park.
You can stay at one of the private reserves and they will drive you around their land. They generally offer a more bespoke experience as far as food and accommodation than what you can get at the national park rest camps, but we think the ability to traverse a wider area with very different ecological environments outweighs some of the extra comforts. The one benefit, though, is that they are not confined to roads or trails on their land, so they can approach wildlife a bit closer in certain cases than you might be able to in the national park.
For example, rhino sightings now are generally confined to the southern area of the park, while lions more generally favor the wider open expanses of the middle part of the park. If you're confined to one of those regions, your spotting opportunities are fewer. Besides, the accommodation and food at the national park was hardly roughing it, and we spent most of our time out seeing animals and scenery away from the camps anyway.
You can also hire a private guide to take you through the park, but we don't think that's necessary either. You are more limited on the hours you can be out and on where you can travel while on drives with a private guide, and they can't offer the walking safari experiences you can do from the park rangers.
Also, we found that having more eyes on the bush meant the likelier you will spot what's out there. Yes, you always have a professional guide/spotter with you, but as good as they are, they can only look where they are looking at any given time. With a private guide, it may be just you and your guide. With the park drives, you have 18 people in your vehicle all spotting.
Once you find something, the guides are very good about making sure everyone can see and get their photo so we never felt like we were crowded out by the groups on our vehicles, and a couple of times, it was our fellow travelers who picked up on something to see.