The dish is a very new consumer tech so we don’t know if its actually secure, but considering the guys behind the product make rockets for a living, the culture in the company should be grounded in compliance and transparency in vulnerability disclosure.
But of course the “first responders” will know first aka military-industrial-intelligence-complex.
Location
The most obvious privacy invasion is of course they know your location, that doesn’t mean they will, in all instances use it for nefarious purposes (know the difference!) - but they would, if the (secret) courts make them.
The Satellites divides the globe into hexagonal zones, so they will always know your location approximately within a 15 mile (~24 km) radius.
They don’t care about the location specifics, only that the plan matches (in-motion, mobile, across country, maritime / ocean, RV etc) FAIR, it costs to run the infra.
Privacy when Buying the product
At least in USA, the site doesn’t ask you any government ID’s while purchasing the kit. As its new, there are not a lot of used kits in the market.
The starlink account is linked to an email with a card to an address that is serviceable. They don’t care if the address you are using it in, is near the address of purchase so just use any (alias) address that you can get the kit off with any alias card, and well an alias email. ( you know the drill )
So at least the dish isn’t linked to you legally. (The next best option is to buy a used kit but the seller needs to transfer the account & that may require some kind of KYC - no idea about the specific logistics)
The basic diagram goes something like this:
Low Orbit Satellite Network (coordinated via GPS) > Starlink’s dish > Router > Switch > your Devices.
It goes without saying having your own router will give you a lot of control, I recommend Mudi v2 as I personally use it, harden it with blue-merle. You should have your own openwrt config, this guy gives you lot of cool options.
The dish has a GPS chip in it and its not just there to be privacy invasive, it serves a function! (the company most likely isn’t nefarious - but it depends on your threat model if you want to trust it, I may trust it)
You are of course better off, switching off the GPS chip, to make the location less accurate.
Hopping
The best way to hide your location is using your own set of devices then changing the dishes you use exactly hence increasing the radius.