I'm more interested in Doppler and timing advance. There's only limited Doppler correction in LTE - the satellites must be compensating for it. Although actually, I suppose they're mostly moving perpendicular to the direction of signal.
Think more “you can reliably send a a short text message” than “you can quickly load up yahoo.com”
It’s all a question of data rates, transmit power, encoding, and (mostly in the geostationary case) large antennas on the satellite end.
That part is answered on the website:
> Direct to Cell works with existing LTE phones wherever you can see the sky. No changes to hardware, firmware, or special apps are required, providing seamless access to text, voice, and data.
It's also notable that this service isn't available yet, it starts (optimistically) sometime in 2024, probably very late 2024.
There, the answers are:
> What frequencies does this operate at?
1-2 GHz. These are also common terrestrial mobile communication frequencies.
> How well matched are the antennas in most phones?
That I don't know for Iridium (I think they're at least somewhat specialized), but for Globalstar, Apple is able to pull it off with their newer iPhones.
> How much more power are you using to be able to do this?
Iridium uses around 1-2 W – that's significantly more than LTE (I believe most devices only use about 125 mW, but I wasn't able to find precise numbers), but about the same as GSM/2G, which current phones largely still support (although I'm not sure whether they use that much transmit power practically still).