The real answer to this is: It depends!
There are limits to some personal information like medical issues. The command may have the right to know certain information about you medically as it impacts your ability to perform but other than that your medical situation is very private and protected by certain laws.
Then there are situations that your Leadership can certainly become involved, should the Soldier allow the issue to enter the workplace. Some examples might include: drugs, alcohol, domestic issues, affairs, inappropriate relationships, etc. These items are punishable under the UCMJ and when brought to the attention of leadership they must take appropriate action.