When cooking (for example, in a restaurant), is it common for cooks in the same team to taste each other's dishes (still in progress), without asking for a permission?
In a restaurant, yes, cooks do a lot of tasting of others work. This is mainly for quality control and to maintain consistency of product. Example: the morning cook's souffle needs to taste like the evening cook's souffle.
Restaurant cooks are team oriented and focused on getting product onto customer tables. Restaurant cooks also learn from each other---most cooks learn On-the-Job, as opposed to culinary school.
In a competition, on the other hand, cooks rarely taste each other's work (with permission or not) before judging.