The short answer
Sometimes, and you cannot find out from Tesla. Free unlimited Supercharging was granted in two forms: tied to the car (code SC01) or to the first owner only (code SC05). Only the first was ever meant to follow the car to a new owner. Tesla's app shows free Supercharging the same way for both, without saying whether it transfers, so a seller can honestly believe theirs transfers when it does not.
What Tesla says to buyers
Tesla’s own support page, under “How do I see if a vehicle I’m purchasing from a third party has free Supercharging?”, answers: “If you are not the vehicle’s owner or a registered driver, we cannot provide that information.” The one party that knows will not tell you, and the seller is not neutral.
How to check
The $1.99 Factory Report on this site reads Tesla's record for the VIN. If Tesla's code for the car is known, the report shows it, with the date it was read. SC01 means free Supercharging was granted to the car. SC05 means it was granted to the first owner only and ends when they sell. If the code is not known, the report gives the odds that the car has it. To know for sure, ask the seller for a screenshot of the Specs tab in their Tesla app, with the VIN showing. If it lists "Free Unlimited Supercharging", the car has it. The Specs tab shows first-owner free Supercharging the same way. For a car ordered around January 2017, use the in-person check below.
In person, there is one more check. On the seller's laptop, open the car's page in their Tesla account, then open the car's picture in a new tab. If the new tab's web address lists SC01, free Supercharging is tied to the car.
The caveat that matters
Free Supercharging is a benefit granted by Tesla under Tesla's own terms. It can be modified or withdrawn at Tesla's discretion at any time, including when a vehicle passes back through Tesla, for example as a trade-in or a used-inventory sale. A code such as SC01 on Tesla's record is evidence of what Tesla granted, not a guarantee that it is active today. Tesla confirms the current status only to the owner, so ask the seller to show it in their Tesla app before relying on it in a purchase.
Which cars can have it at all
Free unlimited Supercharging that goes with the car was a launch-era benefit, on Model S and X ordered by mid-January 2017. Later grants, through referrals and promotions, have usually been for the first owner only. Cars ordered after early 2017 usually carry SC04, pay-per-use. If the car is newer than that and the listing promises free Supercharging, expect it to end when the car is sold.
Common questions
Does the seller's Tesla app show whether free Supercharging transfers?
No. Tesla's app shows free Supercharging the same way for SC01 and SC05, without saying whether it transfers, so a seller can honestly believe theirs transfers when it does not.
How can I check in person whether free Supercharging is tied to the car?
On the seller's laptop, open the car's page in their Tesla account, then open the car's picture in a new tab. If the new tab's web address lists SC01, free Supercharging is tied to the car.
Can Tesla take free Supercharging away from a used car?
Yes, at Tesla's discretion at any time, including when a vehicle passes back through Tesla, for example as a trade-in or a used-inventory sale. Tesla confirms the current status only to the owner, so ask the seller to show it in their Tesla app before relying on it in a purchase.