A ship at extreme distances from earth would communicate using the Deep Space Network, or something very much like it, which is an array of radio dishes spaced out across the globe 120 degrees apart. These communication stations broadcast in a designated set of frequencies (x and ka band) between at 20kw and 400kw of power.
The operating speeds for communication are from as high as is 150Mbps to as low as 10bps. Near Earth the communications are quicker the further out you get the slower the communications are. The New horizons probe communicates at 1kbit per second, which is plenty fast to send text information. A single picture takes several minutes so video with current technology is out of the question.
The good news is that there really is no limit to the maximum range that signals could be received, as long as they are being broadcast with enough power. This Document states that the DSN can pick up signals with a strength of just a billionth of a billionth of a watt of power. This puts the minimum signal power of $1\times10^{-25}w$
Now we can calculate the max distance that a signal could be received from using the Power Density Equation
$P_D=\frac{PG}{4\pi d^2}$
Where:
$P_D$ power density
$P$ is transmitter power
$G$ Gain, for new horizons it is 16000 so we will use that.
$d$ is the distance from the transmitter
So now we can make a guess at the max range a signal could be received for a given power output by solving the following for $d$ with New Horizons gain and a generous 100w transmitter:
$1\times10^{-25}w/m^2 = \frac{16,000\times100w}{4\pi d^2}$
Gives us a detectable range of $1.1\times10^{15} m$ or 8 thousand astronomical units or 47 light days.
I'm not actually an EE so if there is an error let me know. I also know that power density is not a 1:1 for being able to detect something, but this is a ballpark. *Also assuming lossless.