For background, the prison housed the seven men sentenced to prison by the International Military Tribunal. Those sentenced by US-administered courts were sent to Landsberg prison; the British had Werl, the French Wittlich, and the Soviets used various prisons and prison camps in the USSR. Of the Spandau seven, three - Rudolf Hess, Erich Raeder, and Walther Funk - were sentenced to life imprisonment, the other four - Karl Doenitz, Constantin von Neurath, Albert Speer, and Baldur von Schirach - to ten, fifteen, twenty, and twenty years respectively. The prison was jointly run by the USSR, USA, France, and UK, which itself sounds like a sitcom plot. Somehow, they (and the Berlin Air Safety Centre, the only other Four-Power organization that survived the onset of the Cold War) managed to keep going despite the many crises of the postwar period.

At the beginning, conditions were fairly harsh, though with the caveat that the vast majority of Germans in 1947 were living in conditions that were just as bad or worse. While rationing was in effect, they received normal rations, but were not allowed to have their relatives send in food (regular prisoners had that option). At the beginning, they were prohibited from speaking to each other, but that rule stopped being enforced very quickly. They were housed in individual cells but were let out to exercise and work in the yard, which was used as a sort of garden. Communication with the outside world was also fairly restricted at the onset. They could not read newspapers, they could send one letter a month and have one person visit for fifteen minutes every two month. Later, they were allowed newspapers (albeit censored) and more communication.

By the early 1950s, the inmates could socialize quite a bit, the bigger issue was whether they wanted to. As you bring up, Raeder and Doenitz spent a lot of time together but also didn't get along, each one blaming the other's naval strategy for the defeat. Hess got on everyone's nerves with his odd behaviour and claims to have lost his memory.

Overall, it was definitely a 'proper prison' until the 1970s (when Hess was the sole inmate until his 1987 suicide and lived in better conditions than most people his age), but with the caveat that with only seven inmates at the onset, there was far more interaction between them and the guards and staff which allowed for the sort of contacts that normally just don't happen. For example, they all had people smuggling letters to and from prison. The food, once rationing ended, was far better than standard prison fare, and they had access to top-notch healthcare (none of the Four Powers wanted their infamous prisoners to die under their watch). Three of the inmates - Raeder, Neurath, and Funk - were released early because of their declining health, the others served their sentences to the day, unlike practically all of the other criminals sentenced by the four powers individually.

I'm not sure what exactly you'd be interested in, so I'll leave you with a few older comments I've made on the topic. Feel free to ask followup questions if something isn't addressed.

Was Albert Speer truly sorry, or was he just trying to save his image?

Why did Rudolf Hess kill himself?

Why was Rudolf Hess kept in Spandau until his death?

same question as above basically

Was Rudolf Hess murdered in prison? (the answer is no)

Why were the Breda Four kept in prison for so long? I thought you might be interested in another case of Nazis remaining behind bars for a really long time. These four were kept in a regular prison.

References:

Tales from Spandau by Norman Goda. If you're intrigued by the story, this is the best place to start. The book is available on archive.org for free.

Spandau: The Secret Diaries by Albert Speer. This is not something to start with, but once you've got an understanding of the personalities involved and why the author is not to be trusted, it's a fun read full of absurd anecdotes.