Hey @Vítek_Prchal,
I can kind of understand the concerns you are having 
When we started using Asana in our company years ago and I helped with the set-up and monitoring I was concerned about this too. I am kind of a bit of a freak when it comes to organizing things and want to ensure everything is done properly.
How our system works at the moment:
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Before giving team members access to key projects such as let’s say “Main Marketing Project” I would highly recommend that they get familiar with Asana as much as possible. Asana has many great guides online + on top of that I created specific videos about our Asana set-up and structures since every company handles processes differently.
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I also gave everybody their own project so they can test out various things. Now with the improved “My Task” function, this might not be needed anymore. Yet so far I always felt this has been great for team members, also because one of the main rules everybody knew was that “A task not in Asana does not exist”, same for “A task not added to a project is no properly created task”
Now, why did we put so much emphasis on this? Because if somebody ads a task only to their task list and it does not exist in any project then nobody else was able to see the task, except for the task followers.
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Whoever joins our team has a coordinator by their side. Not only to provide guidance for their tasks but also Asana guidance. Now, these coordinators are trained for everything related to Asana and if they have any concerns or questions they would come to me in our case.
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The coordinators would then have notifications enabled for all the projects they are responsible for.
Reason: This way for every single task that is added to Asana they can make sure:
-that the task title is proper (we don’t want short task titles such as “attend to sheet”, a task title has
to explain what it is about in order to also locate it super easy via search, especially when other team members are trying to locate something or checking to see whether a task with the same topic exists already)
-that the task is added to the correct project
-that the task has a detailed description of what needs to be done
-that the task has the relevant followers added
-that the task is assigned and has a due date added as well
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When the coordinators spot something that should be changed they would update the task accordingly and notify the relevant team member.
This is done as long as the team member is familiar with everything and detailed monitoring is no longer needed
Then the coordinator would disable the notifications for some projects but leave them on for to key projects they are monitoring and responsible for. -
Then for certain processes, we have task leaders who report to the coordinator. For these tasks, the task leader is responsible for monitoring and updating accordingly. They would also monitor other team members involved to ensure Asana is used accordingly.
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I am also sending monthly newsletters to our team with updates covering our projects and also Asana tips and tricks and new features.
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On top of that, I am also in a group chat with all coordinators where I regularly share reminders, usage tips related to coordination which include Asana as well. So let’s say I spot a serious error I would point that out there and I would do the same if I spot a really well set up task. This way they get constant reminders which are important to ensure things are running smoothly
So far this structure has worked out quite well to be honest. Sometimes I would just do some random checks to ensure everything is okay.
Also in Asana you can always see when somebody changes the due date, description, etc so in the worst case you can change things back.
Hope that helps