Consider which timeline owns the message as it transits
When does the "message I'm sending to the past" become a "message I received from the future"? Until someone actually does this we'll never know, but my hypothesis until someone proves otherwise: when you link two things across time such that they can communicate, the behavior of the message in transit is what determines paradox resolution.
Basically, it's one of four cases:
Ownership cleany switches as it transits
"Digital" and owned by some other timeline while in transit
(A 3rd timeline, so speak).
"Analogue". Ownership is a spectrum
A message in transit morphs ownership from sender to destination.
(Or - no ownership transfer. This is a possibility, it means that anything arriving from the future is owned by the future timeline in the present, and since it's existence changes the future, destroying the sending timeline, no message from the future can exist in the present. I'm ignoring this possibility as it means past messaging isn't possible, but we're assuming this is possible as part of the question.)
When the message arrives back 10 years ago, the receiver starts modifying their actions, creating a new timeline that may not result in the message being sent 10 years later, in which case, any evidence of the messages existence is immediately wiped from the now non-existent timeline. This results in one of 3 cases:
Atomic. The message appears to of been created out of thin air. (The thought just pops into your head randomly, you have no idea where it came from):
Digital. The message comes out of the medium it would've been put into. (A thought pops into your head that you know came from the future):
Analogue. The message morphs into existence like the timey wimey wibbly wobbly stuff it is. (The thought slowly forms in your head, and like a dream is hard to grasp and remember unless written down immediately)
All 3 of these result in past you being able to build a time-sending machine 10 years earlier, despite the sender no longer existing to send the message.
We don't know how the signal is sent between timelines, but I don't think the medium actually matters here. The message has to change which "time" it's in - regardless of the method of sending information back in time: If you're putting letters in a magic mailbox, or sending packets over an ethernet cable dragged through a wormhole, or linking two minds neuron to neuron across time, or throwing a blood-stained letter through the stargate as it wraps around the sun while it's flairing, that message still changes which timeline its in, and that determines ownership of the message.