It really depends on how you have your Azure DevOps Migration Tools run configured.
The account that you run the migration under will need to have permission to everything that you're accessing.
For example, if you are just running the WorkItemMigrationConfig and migrating work items only (not Area Paths, Iterations, or Tests), you could get away with only having contribute permissions within the destination project and reader permissions within the source.
However, I'd recommend making sure that the account you're using has at least Project Administrator rights within both the source and destination repositories to avoid any issues.
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