In Russia it works this way: you go out on the street, raise your hand and you get your car in like 1 minute (even at 4am in the morning it doesn't take very long time, provided you are downtown, did that many times) - basically some guy who wants an extra buck will stop and get you anywhere. Some make a living that way. You negotiate the price, it's usually rather cheap even in large cities compared to what you'd be charged for the same ride in Europe or the US. You don't have to tip the driver, since the price is negotiated beforehand. Now we also have a lot of competing official taxi companies. They all are relatively cheap and they mostly get their clients through internet or phone and the use case for those is when you need to go to the airport or have a pre-arranged trip.
We also have jitney transportation which is only slightly more expensive, but much faster than public transportation. Drivers are mostly from central asian republics or southern republics - demographics russians are rather hostile to - yet I haven't met more friendly drivers in the whole world. And, in my personal view, people are rather receptive to that and are quite polite with them too. It's pure competition and free market and it works to the benefit of all.
You can criticize Russia all you want, but in some areas the free market here exists relatively untouched (probably thanks to bribes, but still, it's better than regulation).