This is Stephan, protocol designer at Trestor.
There are no transaction fees because there is no security in ensuring that the digital ledger stays the same. What would you do if law enforcement told you to reverse a transaction since 'terrorists' owned trests and were using them?
When you skip that big waste of energy you refer to, there are tradeoffs. I'll pass thanks.
In our protocol, a large majority of Validator nodes is required to agree in order to change the ledger. A law enforcement agency or any other powerful entity would be required to either force a lot of node operators from various jurisdictions to cooperate, or to set up a large number of their own nodes (which would require a large amount of deposits and will not stay undetected).
Trestor does not require fees, because running a Validator node does not require nearly as many resources as running a Bitcoin miner. The idea is, that these nodes are voluntarily run by enthusiasts and organizations (e.g. universities), like we all know it for Tor nodes. The incentives to run a node are of course very different from the incentives why people mine Bitcoins, but that does not change anything about the fact that they are run in a decentralized way by a variety of parties from different jurisdictions.
The hardest problem is to bootstrap the whole network. In the beginning, when the number of Validator nodes is small, the deposit required to run a node has to be set high in order to effectively secure the network. Our plan to get the network started is that the Trestor Foundation will provide organizations, which are willing to run a node, with the required deposit amount.
A reward-based proof-of-stake system is very hard to secure, because there is no way to prevent stake-owners to sign multiple conflicting ledgers with their stake. Some even say that reward-based proof-of-stake systems are completely impossible. We are not doing anything like that. In our system, fees would make no sense, because there is no competition among Validator nodes like we know it from miners. Who is supposed to get the fees, after a majority agreed on a new ledger update?