There are several ways to set up shared BLAST databases for your organisation depending on what you have available. They are:
1. Geneious "Custom BLAST" on a network drive
As described in the post How can I BLAST to a local copy of the NCBI nr or nt databases, a local copy of the BLAST executables can easily be set up to search a local copy of any databases you need. This approach can be used to create shared databases by simply putting the BLAST executables and associated data on a network drive.
Each user can then follow the same instructions for setting up Custom BLAST to access the databases. Their CPU will do the work but that data will be centralised. It is possible that this could cause performance issues over the network though and you'll need to deal with ownership and ensure that your users don't try adding databases themselves.
It is also possible to create BLAST databases outside of Geneious, by running the BLAST+ executable from the command line and using the makeblastdb command. Then put the formatted BLAST database files into the BLAST/data folder where your other custom BLAST databases are located and Geneious users will be able to see them when they run custom BLAST. Note that if the database is formatted manually using makeblastdb, there will be no annotations on the resulting alignments. If it is formatted from within Geneious, then an extra file is created with the annotations so Geneious can put them back onto the alignments after a search.
2. Raw database files on a network drive
Another approach is to provide users with a set of sequences in FASTA format that they can create a Custom BLAST database from on their own machine. You can update these as required but you'll then need to ensure that users recreate their databases with each update.
3. Cloud BLAST
NCBI provides a BLAST server image hosted on Amazon Web Services (AWS). To use it, you must enable it on your own AWS account. To connect to your cloud BLAST implementation from Geneious, go to Tools → Add/Remove Databases → Set Up BLAST Services. Choose Add alternative BLAST server under the Services drop-down menu at the top of the dialog, and enter the address of your AWS instance.
This link provides instructions for setting up Cloud BLAST. Cloud BLAST includes some of the NCBI databases (see here for a list). You can also upload your own preformatted BLAST databases created with command-line blast+.
4. Geneious Server BLAST
Geneious Server is a software package that extends the capability of Geneious to take advantage of your existing enterprise level infrastructure. It comes at an additional cost and includes the ability to create and manage shared BLAST databases.