As redshift occurs due to Doppler effect and universe expansion, how do we know what part of the shift is responsible for which, in order to determine the distance to the object emitting the waves? Is it possible to determine the distance using solely redshift, or do we need some other data as well?
I am aware of other means of finding out the distance to faraway objects. My question tackles the (red/blue)-shift way only as a follow-up question to this Kitchi's answer.
The redshift of very distant galaxies is mainly due to the expansion of space whilst the light has been travelling towards us. The basic relationship (at non-relativistic speeds) is that $v = H_0 d$ where $v$ is the velocity implied by the redshift and $d$ is the distance. The constant of proportionality $H_{0} \sim 70$ km/s per Mpc. That is, a Galaxy increases its apparent velocity with respect to us by 70 km/s for every Mpc (about 3.1 million light years) it is away.
Galaxies also have a "peculiar" velocity with respect to the co-moving cosmological rest frame. This produces a regular doppler shift, but this is indistinguishable in a spectrum from the cosmological redshift due to the expansion of the universe. Typical peculiar velocities, which are caused by relatively local galaxy-galaxy gravitational interactions or motion within a galaxy cluster, are a few hundred km/s.
An example helps. There are galaxies gravitationally bound in a cluster of galaxies that is a billion pc away. The average redshift will look like the galaxies are receding at around 70,000 km/s. If the peculiar velocities are only a few hundred km/s then the redshift is completely dominated by this cosmological redshift.
However, if the galaxies are taken individually, some have redshifts a little bigger and some a little smaller (by a fraction of a percent). This is because the galaxies have their own peculiar motion with respect to the cluster (and this can be measured)
If peculiar motions tend to produce redshifts equivalent to a few hundred km/s, then this is ten times smaller than the cosmological redshift once we get to distances $>40$ Mpc (around 130 million light years). So beyond this, individual galaxy redshifts yield distances to accuracies better than 10% and obviously this improves as we go to greater distance. If you look at a cluster of galaxies then you can also improve matters by $\sqrt{n}$ by averaging over the individual galaxy peculiar motions.
