They're using a different system called "proof of stake".
In proof of work, the more work you put in solving hashes the more chance you have of writing the next block of transactions into the block chain (and thereby receiving a reward).
The idea is that if there are more honest miners(people who perform this computation) vs attackers, then the honest miners will always have solved more and harder to compute hashes then the attackers, so any long enough string of blocks the attackers put forth will not have as much work put into them as a corresponding string of blocks by honest miners.
But this is why its energy inefficient, because people compete to put more work in solving hashes, which has no real useful purpose but takes a lot of electricity.
In proof of stake, the greater amount of in-block chain currency you own gives you a greater chance of writing into the block chain and receiving a reward. So it eliminates the work bit, and hence the gobs of electricity people put into it.
There have been various theoretical attacks against proof of stake, and several technological improvements to it to thwart possible attacks. To date, though, no one has been able to successfully attack any major proof of stake currency.
So to answer your question, the energy necessary to run a proof of stake currency is negligible in comparison to a proof of work currency, such as bitcoin.
In proof of work, the more work you put in solving hashes the more chance you have of writing the next block of transactions into the block chain (and thereby receiving a reward).
The idea is that if there are more honest miners(people who perform this computation) vs attackers, then the honest miners will always have solved more and harder to compute hashes then the attackers, so any long enough string of blocks the attackers put forth will not have as much work put into them as a corresponding string of blocks by honest miners.
But this is why its energy inefficient, because people compete to put more work in solving hashes, which has no real useful purpose but takes a lot of electricity.
In proof of stake, the greater amount of in-block chain currency you own gives you a greater chance of writing into the block chain and receiving a reward. So it eliminates the work bit, and hence the gobs of electricity people put into it.
There have been various theoretical attacks against proof of stake, and several technological improvements to it to thwart possible attacks. To date, though, no one has been able to successfully attack any major proof of stake currency.
So to answer your question, the energy necessary to run a proof of stake currency is negligible in comparison to a proof of work currency, such as bitcoin.