If a class has a method that is private but you want to do a similar action in another class, how would I do this without changing private to public?
Inheritance is the best way to achieve this behaviour by having a sub class extend that class and changing it from private to protected as ttamasu suggested.
However, if they really cannot be compared/contrasted then you could copy the contents of it across. But it is most likely that those two classes will have some common connections/traits that either mean they will share the same abstract super class, or that one will extend the other.