Assuming that the trigger only instantiates classes that are marked as
without sharingdoes that trigger fire in the system context for:
- A standard object page
- A VF page with a controller or extension marked as
The trigger runs in system context for both cases, the with sharing only impacts record visibility, i.e. FLS etc will be ignored in both cases
If a class is not declared as either with or without sharing, the
current sharing rules remain in effect. This means that if the class
is called by a class that has sharing enforced, then sharing is
enforced for the called class.
I believe StandardController references are still an exception and still operate in user mode
stdController.save() in a without sharing controller extension would still error, if the user did not having Edit access on the record.