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Hosting it on a server will make your solution available for others to use if they have proper credentials.
Two people cannot edit the same record at the same time. FileMaker locks a record when someone starts to edit it, and that prevents others from editing the same record. That record could be viewed, but it cannot be edited.
Most developers plan for this: If person A wants to edit a record, the developer takes the user to an edit screen, where all fields are available for editing. During this navigation, the record is opened specifically.
At the end of the edit, a button that says "submit" commits the record, saving the changes. Then it is available for use by other people.
It is easy for a user to forget to click out of a field after changing something. By staying in the field and walking away, they are keeping the record open, which is why you need some sort of 'submit' and commit mechanism.
If Person B wants to edit the same record as person A, they too would try to go to the 'edit' screen, but the script would see that someone else is editing this record, and alert the user that they cannot edit the record because someone else is working on it.
Here's a good link about some of this: Lock FileMaker Records Open