Can I use a t-mobile contract free iphone 6 in europe with a sim card from a local provider? Or: How do I know if my country has a compatible network?
The phone will work (all the typical phone functions) without any problems. From what I've heard, it is only unlocked if you buy the T-Mobile version directly from the Apple store (not from a T-Mobile store). 

As mentioned above, the LTE functionality is the question. Apple's website details the T-Mobile version's included LTE bands (model A1549 (GSM)).

You can check those bands versus what bands your carrier uses to determine if it will give you the full LTE functionality. Wikipedia has an extensive LTE band list per country.