Perder el miedo a hablar otra lengua

[ROBINDESBOIS]

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

[poli]

To lose 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.

[Don José]

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"?

[poli]

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

[Perikles]

Yes, you could also use inhibition instead of fear.