The decision to organize a private Kilimanjaro tour depends upon your priorities, budget and requirements. If having full flexibility over your schedule is important to you, then a private tour may be your best option.
Similarly, if you prefer to go at your own pace and you are concerned about having to adjust your walking pace to match other people, you may also benefit from a private tour.
However, if it is important for you to be able to save money or socialize with other climbers, then you may instead want to join a group tour. Ultimately, the decision to book a private tour will depend upon your preferences.