What is correct? Is there any grammatical difference between them?
I have just stumbpled upon verbs ending with "-se". Before that I was thinking that "voy a" is the only way to say "I'm going to..."
Is "me voy a" fundamentally different from "voy a"?
If you check the ir verb in the dictionary, you will see that there is one fundamental meaning as a pronominal verb:
Moverse de un lugar hacia otro apartado de la persona que habla.
That is, "moving from the current place to another place". So this means that when you say "me voy a dormir", you mean that you are going to move yourself to another place and then sleep. The ir verb has another non-pronominal meaning:
Disponerse para la acción del verbo con que se junta.
This is, "prepare yourself for the action caused by the following verb". You will say "voy a dormir" if you are already in the place where you are going to sleep.
The problem with these sentences is that dormir can also be used as a pronominal verb ("dormirse"), so the first "me" could go with "ir" or with "dormir", so in the end both sentences end up meaning the same thing.
Besides the difference between irse a as expression of movement and ir a as a periphrastic future tense, there's another nuance that I discussed a bit in this question about reír as a pronominal verb.
I would say in fact that Me voy a dormir doesn't sound to me as if the movement (me towards my bed, presumably) is emphasized. Indeed you might say Me voy a dormir even if you're already sitting on your bed. Or you might say Me voy a casa a dormir and there you have dormir as intent instead of movement (a dormir here means the same as para dormir).
What I'm suggesting is that the pronominal element in irse a dormir is not really so much part of irse as a verb of movement, but the same element that appears in dormirse.
Vámonos que me duermo. ("Let's go, [because] I'm getting asleep.")
Me dormí en el trabajo. ("I fell asleep at work.")
Esta noche duermo solo. ("Tonight I sleep alone.")
Dormí como un bebé. ("I slept like a baby.")
There are quite a few verbs where the pronominal form is used for mediopassive meanings (actions that affect the speaker) while the simple form has a more general, neutral meaning. (Dormir just means "to sleep", i. e. to sleep in a certain way, in a given place, for a certain amount of time, etc.; dormirse means "to fall asleep", "to get sleepy", "to get [oneself] a given quality of sleep").
In this same sense, Voy a dormir just asserts that you're going (somewhere, sometime) to sleep, while Me voy a dormir is rather like saying you're going away from other people into your own, personal, sleepy bubble.
"Me voy a dormir" and "Voy a dormir" mean the same thing, and the two sentences are commonly used. I'm from Spain and my native language is Spanish.
Grammatically they are the same, but the second sentence omits the subject (It's very common to omit the subject when you speak Spanish, it's grammatically allowed).
Me voy a dormir. -> I'm going to sleep. (Me) Voy a dormir. -> I'm going to sleep.
I use the two sentences indistinctly.
Existe una diferencia... una dice "Me voy ...", es decir, no es en el mismo sitio dónde se expresa.... La otra dice "Voy a dormir"...
Precisamente, al estar ya en la mesa del restaurante o en el comedor no se ve bien decir "Me voy a comer" pues no debe ir a ninguna parte.
Aunque si estamos en la mesa si se puede estar haciendo otra cosa charlar, interactuar con otra persona y decir "Voy a comer"... eso indica que dejaré lo que estoy haciendo para comer.
