Who is leaglly responsible to disspers belonging without a will

In most cases it has t go through probate... in case there are outstanding debts. The probate court would appoint someone. But it helps to know what country and state as laws vary.

No one. Someone needs to be appointed executor by the probate court. Anyone can apply to the court to be executor. If you are worried, take lots of pictures and make lists of possessions. This happens a lot - people walk off with stuff.

And it depends on what kind of assets we are talking about, and their total value.

For small-value estates there is an abbreviated probate procedure in many places.

Without a will simply means that the assets are distributed according to a statutory "intestate" (means "without will") scheme. Someone needs to apply to be the personal representative.