Hello Mark
No, you do not need to install the self-hosted integration runtime on the same server as the MySQL database. You can install it on a separate machine that has network connectivity to the MySQL database. This machine can be an on-premises machine or a virtual machine inside a private network. If you cannot install the self-hosted integration runtime on the client’s server or network, you can use the managed virtual network feature to set up a private endpoint for the self-hosted integration runtime. This will allow you to securely access the on-premises MySQL database from the Azure Data Factory service.
I hope this answers your question.