Disclaimer: MongoDB employee. All opinions are my own.
Even if your team executes perfectly and never runs into this, the biggest problem IMO is that you can't really enforce most of your guarantees w/ any degree of confidence w/o a typed schema. Even if you work within a typed language that perfectly validates all the invariants of your application before storing anything, the second you need to perform work that does not strictly funnel data through your application (i.e. an update query), you are effectively gambling on whether or not those invariants will hold. This kind of "read-modify-write" flow of data doesn't really perform well (or even hold validity) for a lot of common use cases, so in reality you need your database to ensure these things for you.
Also the two deployed apps problem is just a special case of two people interacting w/ a database who aren't working under the same assumptions as to what invariants should hold. That can happen in single code bases, even with a lot of care taken.
The database schema (along with stored procedures, views etc.) is an API and database engines are designed to have multiple concurrent writers. Multiple applications and users needing access to the same data is largely why databases exist in the first place.