It depends on what you did, but I would say that it is not necessary to install on all computers, in general it is only necessary on the computer that will be the server.
Of course, if you have done everything without thinking about it before you may have problems accessing the server and you will have to fix the application configuration. It is one thing to make to run on your machine, another is to do to run remotely. Obviously on the server you will have to install SQL Server, and if you already have a basic database to start running the application, it should be put there.
If you used an extra library to cease SQL Server other than . NET defaults, you will need to distribute it to your installation.
There are cases that the application may need the database in the client as well. Using SQL Server is not recommended there. At least not in its full version. It is better to use the SQL Server Compact which is an embedded data bank in the application. Or better still, use the Sqlite, even because SQL Compact has been discontinued.
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– Maniero