I would never recommend using both tools at the same time. In my experience, even only one of them causes conflicts/collisions from time to time in a big distributed environments(over 5 teams with access to the same DB).
I will also give you a couple of pros and cons when it comes to running these tools in projects with big, distributed teams:
FlyWay:
Cons:
Very strict when it comes to migration files changes. If someone checked in their migration file, it is not recommended to change it. It's like using force push in git in shared projects. Changing a migration(either its content or the file's name) will cause a migration failure on every machine that had the previous version of the file. The whole migration pack will need to be run from scratch. In some cases it can take a lot of time.
Pros:
Well, what was described in Cons will, at the same time, build up a level of discipline. It might be a reasonable restriction when talking about introducing simultaneous changes by several teams to a production-running application.
Liquibase:
Cons:
With a couple of tweaks should allow you to change the existing(applied) migrations. While it can be considered as a benefit, with a lot of people working on the same codebase, extra caution should be exercised. Flexibility comes at a price. It's easier to introduce some "regression" or inconsistency when such "git force push" style of changes is allowed for distributed projects.
Pros: More configurable and more flexible. Might include solutions for NoSql in the future. A couple of useful plugins(hibernate integration, for example).
