I had the search term "form 4473" is it sent to the government appear in my logs. I guess someone is curious so I will try to answer for them.
The answer is: Yes and No.
Generally, no. The law requires that any ATF Form 4473 remain with the dealer that sold the firearm. The ATF is permitted to inspect the forms as part of normal auditing procedures and during criminal investigations. But as long as the dealer remains in business, the forms stay with the dealer for 20 years. I believe that forms older than 20 years can be destroyed. However, the firearms transfer log (aka "Bound Book") maintained by the dealer is never destroyed and generally contains most of the information found on the 4473..
However, should a dealer go out of business, their "Bound Book" and Form 4473s in their possession revert to the ATF for archiving. Under current law, the ATF is NOT permitted to create a database out of these records. It is believed by many that the ATF has done so anyway and is maintaining an illegal, if limited, firearms registry. That is the "Yes" portion of this answer.
In general sales to you from an FFL, you need not have any worry that the forms you filed out are going to the Federal government. State level is a different thing. Many states have their own state transaction forms and record retention requirements vary widely. Some states retain them permanently and some do maintain firearm registries with them. Others retain them for a short period of time (months to a couple of years) and then destroy them. In such states after the state retention period has expired, the only information the state has that you have purchased a firearm lies with the dealer.
Also note that the approval information from NICS background check is only retained for 24 hours and is destroyed. This is likewise required by law and only some limited, random records may be retained for auditing purposes and then likewise destroyed.