you can not trust the generated private key you must trust the centralized company that produced that tool you are using to generate the key unless the source code of it is open and you can verify whether it is running that source or not.
this is impossible to do with ATMs so you will never have the option to make sure. your only option is to either not use their paper wallets or move the funds right away to a secure key you generated.
for hardware wallets it is half possible, because their source code is open and you can review it but you can not (or at least i don't know of any way) verify that the device in your hand is running the same source code you just saw.
Topic: Trusting private key legitimacy (Read 176 times)
For that matter, when I buy a hardware wallet, how can I trust it to have legit private key? Is there way to verify the legitimacy of the private key?
One easy way is to use your hardware wallet with Electrum.
Electrum is a third party software, unrelated to ledger or trezor, and open source. It can be used with any hardware wallet and will instantly recognize your private keys created by the hardwallet
For that matter, when I buy a hardware wallet, how can I trust it to have legit private key? Is there way to verify the legitimacy of the private key?
What do you mean by "legit" private key? That it is only known to you and no one else? Then it's a matter of trusting the software/hardware that generated the key (a key is just a big random number). Or, if it's open source, then you can verify it yourself, though you need to be proficient in programming to do that. But a lot of other people have probably done it, so you can just check if everything is alright with the latest version of a wallet.
What is more important is to check that the wallet (either software or hardware) is genuine and wasn't tampered with. With software you need to check the digital signatures before installing it, with hardware wallet you should follow the vendor instructions to verify integrity.
if you are talking checking the balance I'm sure the ATMs will also provide the public key you can check them in some explorer like blockchain.com just paste it there and it will show your transaction history and balance.
Not every Bitcoin ATM provides a public key with how they want to motivate people to sweep the funds asap. Simply importing the private key into a lightweight client such as Electrum will instantly show the balance.Electrum has a mobile wallet too so you can do it wherever you are in a matter of seconds.
When you import/sweep a private key into a wallet like Bitcoin Core or Electrum, you can verify that it's legitimate.
Electrum is the more likely option because it's much faster than Bitcoin Core when you import a private key. People without a clue about how Core works and what it is will think something is wrong because it takes so long. 
When I use a Bitcoin ATM to buy Bitcoin, some BTC ATMs allow me to print a paper wallet with the private key printed. Now how can I trust that this private key is just generated and nobody else in the world knows the key? For that matter, when I buy a hardware wallet, how can I trust it to have legit private key? Is there way to verify the legitimacy of the private key?
You can't truly verify that the ATM is securely generating keypairs, nor that nobody else knows the private key. You should always sweep private keys in this situation -- spend the bitcoins to an address you yourself have securely generated.
When you import/sweep a private key into a wallet like Bitcoin Core or Electrum, you can verify that it's legitimate.
If you get printed private keys from ATMs you should sweep them to your own wallet for safety. Not recommended to hold the bitcoin longer in printed keys generated from the ATMs I don't know how Bitcoin ATMs safe so for the safety make your own wallet and sweep the printed keys to your own wallet which you know safe.
If you have a hardware wallet you can't import the private key but you can sweep or just send bitcoin from printed keys to hardware wallet. You can generate a transaction in coinb.in with paper wallets and send them to your hardware wallet.
What do you mean by this "Is there way to verify the legitimacy of the private key?" Private keys should always secret(Never tell other people) if you are talking checking the balance I'm sure the ATMs will also provide the public key you can check them in some explorer like blockchain.com just paste it there and it will show your transaction history and balance.
If you have a hardware wallet you can't import the private key but you can sweep or just send bitcoin from printed keys to hardware wallet. You can generate a transaction in coinb.in with paper wallets and send them to your hardware wallet.
What do you mean by this "Is there way to verify the legitimacy of the private key?" Private keys should always secret(Never tell other people) if you are talking checking the balance I'm sure the ATMs will also provide the public key you can check them in some explorer like blockchain.com just paste it there and it will show your transaction history and balance.
That's why every BTC ATM has instructions on how you can claim BTC when buying at them, surely you have to sweep the funds right after you get the printed private keys out of the ATM. If something goes wrong, e.g private keys dont have funds which I never heard of till now, you can report eventually to the company of the ATM.
When I use a Bitcoin ATM to buy Bitcoin, some BTC ATMs allow me to print a paper wallet with the private key printed. Now how can I trust that this private key is just generated and nobody else in the world knows the key? For that matter, when I buy a hardware wallet, how can I trust it to have legit private key? Is there way to verify the legitimacy of the private key?
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