...But why should I stop doing the exercise when movement gets uneven?
Because at that moment the stronger side takes over and helps the weaker side (with barbells). Or with dumbells you want to train the weaker side and just maintain the stronger side, so you stop when form breaks on the weaker side.
A friend told me to do exercise with dumbbells and use heavier dumbbell for that side that is weaker... I'm skeptical about that. ...
At the same time? Or when doing one-armed DB Bench Press? I haven`t heard about that but that doesn`t mean anything... What did he say about sets/reps when using these different dumbbells? Something like doing 8 reps on the weaker side for example and then doing only 8 reps with the lighter dumbbell on the stronger side? This would be a similar way to what I suggested.