One way to figure out if one (or both) of the objects is a black hole, neutron star, white dwarf, or other compact object would be to try to measure its mass. For example, a neutron star and a white dwarf are both compact stellar remnants. However, there is one decisive factor that determines which type of stellar remnant a progenitor star will become: the remnant's mass.
The Chandrasekhar limit, named after Subramanyan Chandrasekhar, is the maximum mass a white dwarf can have. It is about 1.4 solar masses - quite a lot, if you think about it. If the stellar remnant weighs more than this, it will become a neutron star. There is a similar limit for the mass of neutron stars, the Tolman-Oppenheimer-Volkoff limit. Stellar remnants with masses greater than this will become black holes.
So if you know the mass of one of the objects and know it to be a stellar remnant, you should be able to figure out which type it is. How would you measure the mass? Well, you could study the orbit of its companion star to try and determine the stellar remnant's effect on its orbit. Another way would be to study gravitational waves emitted by the system. These waves can only be emitted under certain circumstances - for example, in a system of binary neutron stars - see, for example, the Hulse-Taylor binary system, also known as PSR B1913+16. The power radiated as these waves, as well as the orbital decay, depend on the masses of the objects. While detecting gravitational waves is an incredibly difficult task, there are several detectors planned or already in operation, such as LISA and LIGO.
While there are other ways to differentiate between such stellar remnants, using the mass of the object(s) can be useful.
Also, a neutron star can indeed have another neutron star as a companion - again, see the Hulse-Taylor binary system.