Perder el miedo a hablar otra lengua

Perder el miedo a hablar otra lengua
 

To loose one' s fear to speak another language. maybe?

That's good.
You can also say: to lose the fear of speaking another language.
This is a less formal way, but just as correct gammatically.

I find that interesting.

Is "the fear" less formal than "one's fear"?

"Fear of speaking" less formal than "fear to speak"?

Yes. Lose the fear ... sounds more natural to me, but both ways are OK.

Yes, you could also use inhibition instead of fear.