Expansion of space vs movement within space I understand that some very distant objects (galaxies) are moving away from us at speeds exceeding the speed of light and that such expansion induced speed is not limited by the theory of relativity. However these objects may also be moving within space at speeds limited by relativity. How do we distinguish between these 2 components of speed, following different laws?
 A: The issue arises because you measure speeds relative to other objects. We have a convenient measuring stick in the form of the Cosmic Microwave Background (CMB), which we believe is "at rest" with the universe.
When we measure the wavelength of the CMB as seen from Earth, we see a slight blueshift in one direction and a slight redshift in the other, so we know we are going in "that" direction travelling at about 625 km/s.
Now consider some distant galaxy with a massive redshift. Well the CMB around it also has a massive redshift. Subtracting the two gives us the speed of the galaxy itself.
A: It might also help to recognize that the concept of "distance" in general relativity is much more subtle than the everyday version because it depends on the coordinate system.  GR gives you the instructions for how to get a distance once you have chosen a coordinate system, which means GR is a "metric" theory.  We are not used to this, we think you can use 2D latitude and longitude coordinates, or Cartesian coordinates in 3D, and still get that the distance between New York and London is a particular thing.  But on scales of the whole universe, you can get very different distances depending on your means of coordinatizing the events.  Because of the cosmological principle, however, there is one coordinate system that stands out, called "comoving frame coordinates" that move with the average of the local matter, because in those coordinates the universe seems to be the same everywhere at a given age.  It is that choice of coordinates that parses between "what space is doing" and "what matter is doing within the space."  Whether this is just a convenient language, or represents something deeper, requires the discovery of new laws to determine, but right now the cosmological principle is a kind of organizational principle of convenience.
Note however that the speed limit of special relativity is not subject to these problems, because that speed limit only applies to two objects passing each other at essentially the same place and time.   The global coordinates don't matter in such a local interaction, just like you don't need latitude and longitude coordinates to tell how fast you are passing someone on the highway.
A: In practice you cannot distinguish between 'peculiar velocity' and 'Hubble flow' per se.  There are numerous ways of accounting for both however.  The main factor to consider is that galaxy obey gravitational dynamics in exactly the same was as any other objects (like stars in galaxies, or planets in star systems).  In particular, their (peculiar) velocity depends on the local environment.  If you observe a cluster of galaxies that are associated with each-other, you can see variations in the velocities of the individual galaxies (peculiar velocity) and also an average velocity of the cluster as a whole (Hubble flow, velocity).  Galaxies in the field (i.e. not in clusters) tend to have much lower peculiar velocities.
Similarly, if you look at lots of galaxies---even ones completely unassociated with each-other---their peculiar velocities will tend to average out, because they're in random directions.
A: When you factor out the expansion of space you're left with the peculiar velocity of an object.
