I'm native AmE, and while I agree that the possessive is usually used in English, I'm also inclined to agree with those who feel that it is not absolutely necessary in all cases.
As an example -- A: "How did you do it?" B: "I did it with my bare hands!" A: "You didn't even wear gloves?!?" B: "Nope, did it with bare hands."
I feel that the use of the possessive is, to some extent at least, context dependent. But since the possessive is often omitted in Czech, there could be a reason, which a Czech native may explain, why it would not be appropriate to omit it here. But if it would be okay to omit it, I will add translations that do not use it.