It is my understanding that the Monero trezor firmware uses the private view key.
Monero Trezor has access to all your keys, but it's contained in the device and thus safe to use.
This might lead to some privacy concerns.
There can be no privacy concerns considering how everything works. Only you have access to your keys and the keys never leave the device (Trezor).
I have read that key image can be used to check balance of a paperwallet.
Yes, the key images are used to check the balance of a paper wallet but you still need the fully-synced view-only wallet.
Does this mean that it is possible to develop a firmware for the trezor that will only use transactions key image to check balance?
Kind of, but it wouldn't make any sense as the purpose of Trezor is for more than just checking the balance - to safe keep your private keys and allow you to also spend without ever exposing your private key. You still need a view-only wallet to make sure that a corresponding key image was produced for every output.
Could this be a privacy improvement?
Over what? There is no privacy loss involved in using Trezor. The only risk is that someone gets your view-only wallet from your PC and the wallet password. Then, he would be able to see the incoming outputs, and that's it. This risk is completely unrelated to Trezor, though.
Monero works with 2 key-pairs: view and spend, and each have the private and public counterpart. The purpose of each is as follows:
- Private view key allows the recipient to see which transaction on the blockchain belongs to him. Without it, he's clueless, because all outputs look the same. Monero are only ever sent to one-time destinations and your address never appears on the blockchain. Only with the view key, you can know which one is for you.
- Private spend key allows the recipient to spend an output by signing a transaction. Also, he signs a key image which is used as a way to prevent double-spends. For each signed output, there's one key image, and if you try to spend the same output again, the resulting key image will be the same and will be rejected by the network.
- Public parts make up the monero address, and are needed by the sender to be able to generate an one-time destination for the recipient.
From the above, with the view key, you can only know what you received, but not if it was later spent or not. Sending the same monero back to yourself would make it look like you just doubled your balance which is not true becasue now the originating output has been spent.
To detect if an output has been spent, the key images are used. Each output corresponds to one key image, but to generate it you need the private spend key. For auditing, you skip the transaction signing, and just produce a key image for each output detected using the private view key. Then, anyone can check the balance, if given the private view key and all the signed key images.
Now, how this relates to Trezor?
Trezor is a device which keeps all your keys. The thing is, those keys never leave the device, and that's what makes everything secure. Your hot-wallet resides on your PC, but is view-only. To spend anything, the Trezor needs to sign the transaction and give the signed version back to your PC in order to push it to the network. You never disclose your view key or your view wallet to a 3rd party so I don't understand what you mean by "privacy concerns".