With socialized medicine, yeah, it could happen.
With universal single-payer, you can go to any doctor you want and the government pays for it. It's the same amount of money for each billable item, regardless of who you go to. And we all pay into it, and all draw from it.
But the doctor does not work for the government. If he wants to just take non-govermental money, he can. Like if he was really good at his job and felt he was worth more than the standard government rate.
And of course the government won't cover anything, so there will be a pool of doctors that do uncovered procedures for cash or private insurance.
If a doctor was an ass, nobody would go back to him because doctor choices would be based on the criteria of sucess, skill, trust, and personality. If your doc's an ass, you can pick a new one.
But the records have to be kept private, with the doctor you go to. Nowadays they can easily be transferred to anyplace in the country.
The other possibility is some sort of
public/private key encryption system, where data can encrypted by anybody but only you can decode it.
In this case, the government medical system has the public key. Your doctor can take your files and chart that are on his computer and encrypt them using your public key to a national database. He would be able to look it up the public key and use it. Now, if you needed to see a doctor while on vacation, you would simply give him your name and address (or an ID number), the new doc downloads your files, and you use your private key to decode it in his office. After the visit, he encrypts the visit's new information and uploads it to the national database again.
If you were in an accident, the hospital ER could contact your current doctor on an emergency basis for the unencrypted data stored on your doctor's secure office network, and it could be emailed or faxed immediately.
The national database could act a as a secure 3rd party for verification purposes, to make sure that the public/private key is corrupted or replaced. And it would be stored in a secure facility, encrypted, so that a hacker would first have to hack into the database to get the information, then crack the information he downloaded.
Keeping the information encrypted by your private key in a government database means that the government can't simply browse through your information at will. Aside from the legalities of a court order, they would also need some heavy-duty encryption from the NSA or FBI to crack your files.