For a user to access the BOM module, he/she would need to be given access to BOM under user management. The BOM menu items under Products would appear after this.
You can further restrict access to each BOM by user. Activate this by going to Setup, Entity Feature Setting, BOM, Setup BOM access at the user level and check the checkbox. Save the changes.
Go edit each BOM (Products Menu, BOM) and you will be able to assign selected user to have access to this BOM. User with firm admin access do have access to all BOMs even if the user is not assigned to a BOM. User with no access to a BOM (but have access to the BOM area) will not be able to do a build with this BOM or see past builds for this BOM.
You can also check the setting for Obscure GL posting for Product Build with no line detail description to skip recording details of the build in the GL entries.