The ACC-ing structure may be more appropriate than the POSS-ing structure on occasion, and vice versa.
We watched him leaving the building to see if he remembered to lock up.
We expected his leaving the company to take place long before it actually did.
The variant with him focuses more on the person, the variant with his more on the event. In the above two examples, this strongly suggests, or dictates in the second case, which variant should be used. In the original, I think him (more personal) just shades it. Either is grammatical, though the POSS-ing option is more formal and will often sound pretentious in many registers.
A member mentioned the fact that some educational establishments regarded the ACC-ing construction as ungrammatical in the fairly recent past. The only historical treatment I've found is from Nonfinite Structures in Theory and Change By D. Gary Miller, which includes
Around , NOM-ing (replaced by ACC-ing in about 1900) split off
This implies that school grammars of the mid-20th Century which forbade the ACC-ing structure might be seen as hyperprescriptivist. Which comes as little surprise.