Your neighbour has two roles, one as a "normal citizen" and one as the caretaker of the building, appointed by the agent or owner.
My sense is that the garbage first belongs to you, then later belongs to the refuse collection service. In between, the landlord is housing the garbage, as it were, temporarily, in the container, or on the side of the road on the landlord's property. At no time does the garbage become the property of any neighbour. However, the caretaker may have legitimate reasons to look through it. You could ask her (or the agent/landlord) what they might be.
My tip is to ask your caretaker what, exactly, she would like done with the garbage.
Take your time, and ask whether there is anything you are doing with your garbage, perhaps without realising it, that she thinks you should be doing differently? How does she think the bags should be tied? In which bags? Placed where? Could she walk with you now, and show you the spot? Is there any limit to the number of bags one is allowed to throw away in one week? Are there written rules about this, from the municipality? What are the advantages and disadvantages of using 17 litre vs. 35 litre bags? Does she buy hers from Migros or from Denner? Does she have a list of any relevant times and dates of collection? Could she please fetch such a list from her flat, and you would come with her now, to collect it?
I saw this brilliant tactic work very effectively with the High Priestess of the Shared Laundry.
The man determined to go very, very slowly. He asked her to show him how, exactly, she believed that the washing machine should be cleaned, and to show him which cloth she thought was the right one, where it should be placed after use, or whether it would be preferable if he brought his own cloth for this purpose. He asked whether she herself had a preferred and most efficient sequence of wiping down the machine, tumbler and air-vent, asked her to show him where she usually discarded the textile fluff, with which brush she cleaned the filter, and could she please show him right now, the best way to do it.... etc., etc., oh, and just one more thing, please, ad nauseam . And she was transformed from terrorising him to never, ever bothering to moan about anything he did or did not do, ever again!
https://youtu.be/QxBnaMGP2aY
Just one more thing, please...