Sending waves through the ocean would not be a good option, for many reasons. It would require a significantly higher amount of energy to 'push' a signal through dense water, and there would be far more things that can interfere with the signal causing it to degrade further.
While I think DRBob suggestion is quite interesting, I can't see this being viable for long (cross continental) distances, or frankly any technological communication, though quite viable for an evolved pre-technology communication system for some sort of whale-song communication.
Part of the problem is that for a signal to be strong enough to reach long distance it needs to be so 'loud' that it's harmful to things nearby, 'deafening' everyone around them, if not causing more drastic harm. The energy requirements are also simply too high vs other technological options available.
For long distance comes they will use the same trick we use, Fiberoptics. I know you didn't like this answer because it's recent technology, but fiberoptics is only the most recent approach to large bandwidth wire comms, and more to the point cross-continental communication is also pretty recent. Our very first cross-continental reliable comm used wired communication strung along the seabed and there is no reason to think merfolk wouldn't go the same path, it would be even easier for them then us. It's boring, but obvious approach.
If instead your looking for semi-long distance communication, such as with radio or walki talkie, then, again, they will likely use the same approach as us, radio waves over the air. This is because it takes much less energy to transmit over air and signal won't get corrupted as easily.
You mentioned a concern about crossing the water/air boundary, that isn't the problem with this approach. This boundary makes things harder if we actually tried to transmit signals directly across the water/air boundary; but we wouldn't.
Instead, we would use the 'cell phone' approach of setting up transmitter 'bouy' on the surface. These bouy rest within the water, allowing easy access to merfolk, but would project out of the water into the air. They would receive the merfolk signal, broadcast it as radio waves, then have other buoy pick up the radio wave and transmit it back into the water. Thus allowing the majority of the signal to travel through the much easier to transmit open air.
The bigger issue is, again, the short range that waves can transmit through water (particularly at energy levels cost efficient enough to be worth producing). I imagine Buoy would be 'wired' devices for quite some time, with merfolk traveling to the nearest Buoy and connecting to it to receive signals. It won't be until we have better computer processing ability to allow more complex error correction algorithms, and timeshare algorithms similar to how wireless routers divide up requests form multiple machines, that it will be possible for a Buoy to send 'wireless' messages through the water. In fact a Buoy would likely act almost exactly like a wireless router in how it functioned, though perhaps a more simplified version of one.
The only exception for that is if you have a Buoy designed to receive/send only one signal, which allows that one signal to be broadcast at much higher energy levels through the water. I could see people setting up a sound Buoy that broadcast over very short range that they swam with like personal floating radio for instance.
However, I believe people would very quickly move towards complex wired approach since long range radio comms required one to access a Buoy. With many wireless hotspots that cover only a very short range, almost exactly like what we use today except that the hotspots have to be slightly shorter in distance and it's a little more draining on the battery of your wireless device to use them.