We shouldn't need one antenna per face of a world. At most, we should only require two, each on opposite faces to the other (for shortwave stations).
I've given more thought to this, and I've come up with a solution that should work. Handheld radios should be able to transmit at varying powers, to allow the user to either conserve power or restrict the range at which third parties can hear transmissions (for example, for transmitters in the V/UHF band, a transmitter operating at 2.5w would have a range of 1.5km, 5w would be 3km, 10w would be 6km). Also, line-of-sight between transceivers should increase range / signal quality.
You know what? I'm gonna categorize all this.
HANDHELDS
[*]Shortwave handhelds - able to refract signals around the edges of a world (but not that far), and don't require line-of-sight, but are more inefficient (and therefore use more power).
[*]V/UHF handhelds - more efficient and have higher bandwidth / better signal quality, but are line-of-sight only (except for minor obstructions like walls and trees).
[*]Beam handhelds - a regular handheld of one of the aforementioned types, but has a beam antenna (like a parabolic dish or a Yagi, for example) in place of the regular antenna. The beam antenna has to be aimed straight at the receiving / transmitting station (so you can only communicate with one station at a time), but range and signal strength (and therefore quality) are phenomenal. You could communicate with spacecraft using the V/UHF version, or communicate across a planet using the shortwave version.
BASE STATIONS
[*]Shortwave base stations - can transceive over very long distances, and are pretty efficient, but require a large antenna (20 meters in height for best efficiency, maybe about 6 meters at the very least if you need it to be _really_ compact). Can also use turnstile-mounted Yagi antennae, but said Yagis are very large (if memory serves, 10m-band Yagis are generally about 15 meters long).
[*]V/UHF base stations - are line-of-sight only and use more power, but give higher bandwidth and better signal quality than shortwave. Antennae are much smaller - 2 meters for regular antennae, and 1.5 meters long for Yagi antennae. Are able to use large parabolic dishes for communications with distant spacecraft, or either dishes or Yagis for communication with closer ones.
[*]Microwave base stations - provide the highest bandwidth of all (several times higher than that of V/UHF), but eat up power like nobody's business, and are extremely line-of-sight (a single block between two stations is enough to block all communications). Can only use beam antennae, but those antennae can be mounted in a ring on a pole, as they are very small (half a block in size, at most). Can communicate with spacecraft, but use ungodly amounts of power doing so.
ANTENNA TYPES
[*]Monopoles - basically your regular antenna. Non-beam handheld radios use these. Feedpoint (input) is at the bottom. Radiation signature is strongest perpendicular to the antenna, and weakest at its zenith (parallel to the antenna). Basically what this means is that for a vertically-mounted monopole, the signal is most strongly directed toward the horizon, and most weakly directed up-and-down. Can be mounted / oriented horizontally for communication with stations that aren't directly on the horizon.
[*]Dipoles - Basically two half-length monopoles mounted directly next to each other with their feedpoints facing toward each other. Its radiation signature is stronger perpendicular to the antenna, but even weaker parallel to it (to the point that communications with stations parallel to the antenna is nearly impossible). Can be mounted either vertically or horizontally, just like monopoles.
[*]Yagis - A 'beam' (directional) antenna. Has to be pointed at the receiving / transmitting station to work (or be angled properly at the upper atmosphere to bounce the signal around a planet's edge), but provides unparalleled signal strength. Feedpoint is near the back of the antenna, but not directly at the back.
[*]Parabolic reflectors - basically just a satellite dish with a beam antenna mounted at its focal point.