It sounds like there are multiple issues here.

First, where is the SQL Server service running that has this database, on your computer or on a server?

Second, when you say the database is "on a network drive so it is properly backed up", a file system level backup is not adequate protection for a database.  Please make sure that BACKUP DATABASE command or a third party tool that performs database backups is used on a regular basis instead of file backups.
https://docs.microsoft.com/en-us/sql/t-sql/statements/backup-transact-sql

Third, you say everything "was created locally on my PC", but you say the database file is on a network share.  Was the file created in SQL Server on your computer, detached, then moved to the network share, and then reattached to another SQL Server, or how did this file move?

Fourth, file permissions across a network share are based on the permissions granted through the share and on the server's local file system, not on who originally created the file and where it was created.  Check with an admin to find the permissions for that network folder.

Fifth, when you say you are adding the database to Visual Studio, do you mean you are adding a reference to the database in a Visual Studio project?  Some clarification may be needed here.