Do it if it is useful for your process.
Some teams do not do it because they want the pool of work in a sprint to be genuinely fully handled by their self-organised teams.
Some teams don't want their sprint starting until they know exactly who should be doing what (this is not quite the Scrum principle, as it only really helps in teams where you've got mostly specialists who can't pick up any story and run with it like you should in scrum, and you're allocating stories before so you know not to overload one specialist while leaving the others bored. Works better in Kanban)
A lot of teams tend towards the middle - they are multi-skilled, and in theory any of them could pick up anything, but in real life, especially during sprint planning, and sometimes grooming, an individual will say "oh, I'd like that one", or "I could learn from that", so it gets assigned when they notice.
If there is no assignee to a task, then it depends more on your process. In "proper" Scrum and Kanban you take the highest ranked story from the top of the board and do that next (in Scrum, you do that at the start of the sprint for everyone), and then you take the next highest when you finish your current one. Ideally, when you start work on a Story, it should be assigned to you, so everyone knows who the responsible person is.
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