Can an object access a private member of another object of the same class?

Yes, an object can access private instant members of other objects in the same class. Let's re-exam an example from "Why always override hashcode() if overriding equals()?" in XyzWs Java FAQ.

public class CustomerID {
  private long crmID;
  private int nameSpace;

  public CustomerID(long crmID, int nameSpace) {
    super();
    this.crmID = crmID;
    this.nameSpace = nameSpace;
  }

  public boolean equals(Object obj) {
    //null instanceof Object will always return false
    if (!(obj instanceof CustomerID))
      return false;
    if (obj == this)
      return true;
    return  this.crmID == ((CustomerID) obj).crmID &&
            this.nameSpace == ((CustomerID) obj).nameSpace;
  }

  public static void main(String[] args) {
    Map m = new HashMap();
    m.put(new CustomerID(2345891234L,0),"Jeff Smith");
    System.out.println(m.get(new CustomerID(2345891234L,0)));
  }

}

In the above example code, you can see that crmID and nameSpace are private fields.