What an awesome and insightful question.

Yes, and no.

Let's make a few assumptions. First. Let's assume that both planets are aware of each other and have tech's assumed at each other. Second let's assume that the communication is instantaneous to take out the math issues. Third, only one planet can transmit the messages but the other can recurve them. Finally, assume I'm a random stranger on the Internet and I don't have a PhD in physics.

Both planets are looking at each other and they are seeing photons that bounced off the other planet 4 years in the past. (i.e. they are 4 light years apart) If one planet transmits this instantaneous signal the planet that received the signal will see it as coming from the future because it was transmitted 4 years before the images they are recording on their telescope. So, if they could see the transmitter being built they might see it still under construction for four years after they get the first transmissions.

To the planet transmitting the signal they would get feedback via light speed communications or actual light images in only 4 years rather than the anticipated 8 with communications limited by the speed of light. I think this would look like time dilation rather than time travel to the planet transmitting.

It is not dissimilar to shooting something at long range. If you are the shooter you hear the gun fire and then see the target get hit. If you are standing near the target you see the target react to getting hit long before you hear the gun shot. So, which order you perceive things happening depends on where you are. Of course we are used to the difference in asked between sounds and light so it seems natural but it takes a bit more to shift ourc thinking you you're question