There is no database as such in Bridge as it is a file browser, the metadata is embedded into each image file.
If user A has a specific keyword in their Keywords panel and applies it to an image, then user B views that image but does not have the same keyword in their Keywords panel – then it will be listed in italics under the “Other Keywords” category as a non persistent keyword (right click to convert to a standard keyword). Otherwise users can export and share their updated Keyword panel data between themselves.
In other words, the keyword metadata is common to the file, not the program – however there are multiple (manual) methods to permanently store new keywords entered by other users in each separate program. Bridge is not designed as a multi-user networked DAM tool and requires some work for multiple users.