Is the language
$\qquad L=\{ \langle \text{M} \rangle \; | \; \text{M is a Turing machine that decides some language} \}$
a Turing-recognizable language? I think it's not, as, even if I am able to tell somehow that a Turing machine halts for some input there are still infinite strings to check for. Similarly I think that this problem is not even co-recognizable. Am I right? If yes is there a more precise proof ?
Of course, this depends on what you exactly mean.
Do you mean, all the machines that decides a specific language? e.g., $$ L = \{ \langle M \rangle \mid M \text{ decides the language } A\}$$
then, it depends on the language $A$. For instance, if $A=HP$, the halting problem, then $L$ is clearly decidable (i.e., it is empty).
But if you mean, any language, i.e., that $M$ is a decider, $$ L = \{ \langle M \rangle \mid M \text{ halts on all inputs } \}$$ then $L$ is not recognizable, see Yuval's answer.
This language is usually known as TOT, the language of machines computing total functions. It is $\Pi_2$-complete, and in particular is neither recognizable nor co-recognizable.
