When I am using ASSM and I am doing multiple inserts into the table does each session obtain a different block from the bitmap and insert into it or does all the session insert into the same block?
It's not mandatory that each unique session would get a unique block only for the insertions. If one block still has the ITL slots available, it would be able to serve multiple session's inserts without any issues. The use of ASSM would relieve the contention from the segment's header block as that would be in severe contention when used with the Free lists .
MY understanding of the ASSM bitmap model is that hash of your process ID is used to determine which branch of the bitmap you search, thus eliminating contention between sessions. RAC includes the instance ID in the algorithm, thus eliminating contention between instances. So in an ideal world, your concurrent inserts will go into different blocks.
This was discussed a year ago here, Re: Locally Managed Tablespaces vrs ASSM