If you take a tape of a high pitched singer and play it back slow, the pitch will seem lower and take longer to play back. A similar effect happens here due to regular Newtonian Doppler Shift that cause the pitch to appear lower when going away from the source and when combined with time dilation the effect is exaggerated even more.

The traveller (Betty) going away from Earth considers herself to be stationary and the Earth is going away from her. From her point of view the Earth is time dilated so if the Earth communicator (Bob) sends out beeps at a rate of once per second by his clock, then to Betty it appears as if he is sending out beeps at a rate of once every two seconds (if the relative velocity is such that the gamma factor is 2) and this is on top of the regular apparent slow down of the signal due to regular Doppler shift.

To be more specific, the Newtonian Doppler shift is given by $$F_r = F_s \left(\frac{c+v_r}{c+ v_s} \right) .$$

where $v_r$ and $F_r$ are the velocity and the frequency received by the receiver and $v_s$ and $F_s$ are the velocity and the frequency sent by the sender. In this particular case we will consider the velocity of the receiver to zero and after dividing the top and bottom by c the equation becomes $$F_r = F_s \left(\frac{1}{1+ v_s/c} \right) .$$

Now we divide by the relativistic gamma factor to allow for the reduction of the received received frequency due to the time dilation of the sender: $$F_r = F_s \frac{\sqrt{1-v_s^2/c^2}}{(1+ v_s/c)} = F_s \frac{\sqrt{1-v/c}\ \sqrt{1+v/c}}{(1+ v_s/c)} = F_s \frac{\sqrt{1-v/c} } { \sqrt{1+v/c}} \ .$$

This is the Relativistic Doppler Shift and it is reciprocal. Bob will perceive messages from Betty as frequency shifted downwards by the same factor.

If the resultant shift in the received pitch is too low to be understandable, they would have to record the messages and play the tape back at a compensating increased speed to make the audible message sound normal. On top of this, (as @m4r35n357 mentioned in the comments), if Betty is one light year from Earth, Bob will have to wait at least two years from the time he sends a question to the time he receives an answer from Betty. This delay due to the finite speed of light and large distances is already a problem for communicating and controlling probes sent out into the far reaches of the Solar system.

If you take a tape of a high-pitched singer and play it back slow, the pitch will seem lower and take longer to play back. A similar effect happens here due to regular Newtonian Doppler Shift that cause the pitch to appear lower when going away from the source, and when combined with time dilation the effect is exaggerated even more.

The traveler (Betty) going away from Earth considers herself to be stationary, and the Earth is going away from her. From her point of view the Earth is time dilated so if the Earth communicator (Bob) sends out beeps at a rate of once per second by his clock, then to Betty it appears as if he is sending out beeps at a rate of once every two seconds (if the relative velocity is such that the gamma factor is 2) and this is on top of the regular apparent slow down of the signal due to regular Doppler shift.

To be more specific, the Newtonian Doppler shift is given by $$F_r = F_s \left(\frac{c+v_r}{c+ v_s} \right) .$$

where $v_r$ and $F_r$ are the velocity and the frequency received by the receiver and $v_s$ and $F_s$ are the velocity and the frequency sent by the sender. In this particular case we will consider the velocity of the receiver to zero and after dividing the top and bottom by c the equation becomes $$F_r = F_s \left(\frac{1}{1+ v_s/c} \right) .$$

Now we divide by the relativistic gamma factor to allow for the reduction of the received frequency due to the time dilation of the sender: $$F_r = F_s \frac{\sqrt{1-v_s^2/c^2}}{(1+ v_s/c)} = F_s \frac{\sqrt{1-v/c}\ \sqrt{1+v/c}}{(1+ v_s/c)} = F_s \frac{\sqrt{1-v/c} } { \sqrt{1+v/c}} \ .$$

This is the Relativistic Doppler Shift and it is reciprocal. Bob will perceive messages from Betty as frequency shifted downwards by the same factor.

If the resultant shift in the received pitch is too low to be understandable, they would have to record the messages and play the tape back at a compensating increased speed to make the audible message sound normal. On top of this, (as @m4r35n357 mentioned in the comments), if Betty is one light year from Earth, Bob will have to wait at least two years from the time he sends a question to the time he receives an answer from Betty. This delay due to the finite speed of light and large distances is already a problem for communicating and controlling probes sent out into the far reaches of the Solar system.