At JPL I participated in several studies concerned with telecommunication from depth at Jupiter (and other giant planets). In general, with radio telecom from within Jupiter's atmosphere, the lower you go in frequency, the deeper you can penetrate. But there is a practical limit.
Above about the 100-bar level (the depth in the atmosphere where the pressure is 100 bars) most of the absorption of radio waves is done by ammonia vapor and water vapor. Those molecules have absorption lines from wavelegths of a few tens of cm (not very many there, and they are weak) to the rotational and vibrational lines that are strong and there are oodles of them, at sub-mm and shorter wavelengths. Those strong lines make telecom at short wavelengths impractical at any considerable depth. As pressure increases a given line's linewidth (the band of frequencies over which there is significant absorption) also increases, so each line takes up more of the comm bandwidth, eventually overlapping with others and making large bands unusable. Even far (in frequency) from the line center, a line contributes some absorption, and that contribution increases as pressure increases. At radio wavelengths, orders of magnitude in frequency from the sub-mm and IR line centers, high pressures can cause the "wings" (far tails of an absorption line) of strong sub-mm/IR lines to cause more absorption than the lines within the radio band. Although this reference discusses absorption at optical wavelengths, the same principles apply, though with different absorption mechanisms.
Fortunately, on the low-side tails, the farther you get from the line centers, the weaker the absorption is. So the lower you go in frequency, the less absorption you get. Un-fortunately, the lower you go in frequency, the less bandwidth you have for communications, so lower data rates, video resolution, etc. This is the telecom trade to be made: how deep do you need to go, what data rate do you need, what frequencies are practical (low frequencies mean larger, heavier, and clunkier components like antennas, waveguides, etc.), and how much power do you have to pump into the signal to get it out to the receiver? The more power you need, the bigger and heavier is the power supply.
As you go below the 100-bar level other molecules get involved, such as phosphine, with absorption lines all the way down into the MHz range. Even hydrogen shows some collision-induced (molecular-to-molecule collisions, not collisions with the vehicle!) absorptivity. The atmosphere gets pretty opaque down there.
But it gets even worse. The deeper you get, the hotter it gets. At some point (roughly 1000-1500 K) the combination of temperature and pressure is enough to ionize some of the atmospheric constituents, so the atmosphere becomes highly electrically conductive. It's like trying to transmit through metal! Even with @uh-oh's VLF or ELF.
Before I left JPL to pursue life as a consultant, they were still trying to find practical ways to communicate from the 100-bar level at Jupiter to an overhead relay vehicle — an orbiter, a flyby spacecraft, whatever. I proposed an approach that appeared the most promising: use a staged probe, where one part deploys a big parachute and stays high in the atmosphere while another part goes deep and has to communicate only the relatively short distance to the upper part; then the upper part relays to the in-space vehicle. You can even use three or more stages. The main problem was that Jupiter's strong wind shear causes the upper one to be carried laterally far from where the deeper ones are, making the relay link deteriorate quickly. We thought of ways to tackle that, but they all involved fairly complex systems and were looked at as too risky.
The question and comments have mentioned non-electromagnetic means of telecom. Sound isn't very useful, as @AJN says, because it looks like Jupiter's atmosphere is one giant, complex storm. The noise level would be very high, meaning that to get a useful signal-to-noise ratio ('SNR') you'd have to put a lot of power into the sound. @James suggested gravitational waves, but sometime down the line ... way down the line ... way, way down the line! Let me know when I can buy a gravitational-wave generator of useful output that's smaller and lighter than a neutron star! A receiver, too, much smaller, lighter, and more sensitive than LIGO. Most particle-based schemes (like neutrons) also suffer from the dense atmosphere, as @Jon Custer alludes to: the mean free path ('MFP') in the deep atmosphere is short compared to the propagation pathlength needed to get out of the atmosphere. Neutrinos have much longer MFPs but have some of the same problems as gravitational-wave schemes: practical systems need to be much smaller and lighter than Super-Kamiokande.
The suggestion of relay nodes using hot-air balloons is a version of my staged-probe method. The two main problems are the wind shear, which spreads out the stack quickly, and: you don't get much buoyancy from heating air that's mostly hydrogen. The low molecular mass is a big problem, one that requires heating a large mass of air. The more air you try to heat, the larger (and heavier) the balloon envelope gets, and the more heat you have to supply due to convection and radiation losses. Many years ago I did a related calculation: using 238-Pu (the same isotope they use in NASA RTGs as the heat souce), how big does a Pu-heated jovian balloon have to be to support its own weight and the weight of the Pu? Using the highest strength-to-weight-ratio materials available at the time, it finally converged at a balloon radius of some 30 km, and needing mega-tons of Pu. And that's without any payload!
Communicating into and from planetary interiors has been, and continues to be, a problem that has a lot of smart people still scratching their heads.
