Which of this options is better: replicate a database or share the data?

Hi!
I have to integrate two databases in order to generate a solution suitable for two institutions. Each institution has its own database, one of them is a people register which is the base of the services given for the other institution...  I need to know your opinion about crossing both databases... What do you think is best, to replicate the people register from one database to another, or cross the information between both databases? This solution is very important, so I need to know what the best practices are around this...

Any help will be appreciated...

Beth