Can we change drivers without the rental firm getting to know?

@phoog's answer is very good, but I'm posting this answer to place more emphasis. Do not even consider doing this. Only the people who are named drivers of the vehicle are insured to driving it. This isn't just a matter of you being liable to pay damages if you're in an accident (which could be a huge amount of money depending on what/who you damage/injure/kill). In most jurisdictions driving without car insurance is a serious criminal offence which can see you fined, banned from driving, or even jailed.

There has been mention in the comments of relying on your own car insurance to let you drive other vehicles. This is very unlikely to work, unless you have a particularly lenient policy document. Most will place restrictions on driving in foreign countries and, more importantly, require you to have the consent of the owner of the third party vehicle. In this scenario you would not have their consent.

The legal and boring answer is: you can't. Changing drivers will violate your contract (which is a civil violation) and violate the terms of your car insurance.

The real life answer is: no one will find out in 99% of the cases. If the police stops you, they rarely verify the name on the rental contract and it's even more rare for them to care about a wrong name if the "official" driver is in the passenger seat. Some might check insurance, but you could just buy some cheap insurance in your own name to show to the police.

Theoretically speaking you might get into an accident with another vehicle with plenty of witnesses, which no insurance will agree to cover, but that's just a risk you will have to take. Estimate that risk and make a choice.

I usually rent a car to travel with my family during weekends.

I would guess if someone else drove the car instead of me, the company would never know unless you have some accident. And then you are in serious trouble, because insurance would not cover it - it's in the contract.

It is a bit like those european cities where you buy a public transit ticket, but there is usually no one on the bus or train to take it. You keep it in your pocket. Or you simply walk into the train and ride. A lot of times this will work and you'll save some money, but if the ticket man comes, you are fined.

In the end it would be a matter of risk versus benefit. My personal rule is the following:

Only ask for a second (unauthorized) driver to take the wheel if the risk of accident with the this driver is significantly lower than yours.

This usually means asking my wife to drive if I am too sleepy or tired - which is not uncommon.

But, of course I should not be doing this, because the extra fee is insignificant given the peace of mind. Well, at least in Brazil.

You'll have bigger problems then the car rental company, you wont be insured and will be committing an offense. Besides the face that if you crash then you wont be insured.

If you buy insurance or a loss/damage waiver (LDW) from the rental-car company, it won't cover any damage that occurs while a non-authorized driver is at the wheel, if they find out.

But, don't buy insurance or a loss/damage waiver from the rental-car company. It's a scam.

I was renting at Alamo and the guy was pressuring me to buy LDW to cover my $1000 deductible, for $25 a day (on a $12-a-day car). I said, "Do you think I get in an accident every 40 days?" The guy just gave me a blank look. "$1000 is $25 a day for 40 days. I get in an accident every 10 years." Alamo sucks.

First, see if your own auto-insurance will cover you. In most cases, it will in your own country and won't in other countries.

Next, see if your credit-card will cover you. In most cases, it will. It won't in Italy. Italy sucks.

I rented in France. The rental company (Avis) in Marseilles claimed I scratched the hood and billed me $600 for it. I filed a claim with my credit-card (AmEx). AmEx tried to get some sort of explanation out of Avis for six months, and finally gave up and paid me $500. France sucks. Avis sucks too.