Now until game or
Vanderer clarifies, it could be compared to death sentence, since their being is basically snuffed out of existence, but leave body to work for society.
I intended to answer, but forgot. Better late than never, though.
No, they do not lose their mind. But they have to obey the brand owner orders to the letter. That means, for example, that if the order to protect someone is given, they will do it, but the method will be up to them. Of course, like the MC said, there are ways around the orders, but the punishments are very severe. Death, usually, and not a quick one.
The Condemned are shock troops. Think penal battalions in WWII Russia. They all got the worst assignments of any troops, and the worse equipment. Need a breach in an enemy line during a battle? Send 200 of them. Even if they fail, well, the enemy will be tired at least. Or they are sent to fight the worst monsters, you get the drift.
As for Nenah, the sister who stole, her sentence was lenient. She stole from her Lord, and from orphans. Andolisa, the grandmother, is a minor Deity, but one that is very respected. And you don't steal from your Lord or Lady... If she was not fooled into it, Athagan would have given her the same choice than Alyah: Death or collar.
In general, punishment in Aetas are far harsher than in modern days, but they often give those condemned to death the choice to be collared instead. With the low population, it is more useful to use them than to just kill them outright. Exceptions to this are for child killers, serial murderers, or people able to resist the brand for example.