My son has it...and the result of his is on the more negative side of things and not what most kids with this would face. So I will leave out the unlikely stuff and just answer about speech therapy =)
Our son is two corrected and though his speech has been a 2-3 months behind (if you want to call it that?) he does talk and we see speech therapy like once every three months...so they are not that worried. He knows over 50 words, and can say basic sentences. He makes animal noises and sounds of letters as well. He can even read two words now...sort of scary!
He however did not coo or babble like most infants, he made noise but it wasn't like a normal newborn either. His voice has always been raspy and quiet. Most people understand his speech, some do not. He will likely have a very deep raspy voice, husky if you will when he is older.
I have done SO much research about this topic, as it's our son's only lasting NICU side affect and it seems like most kids (not ours though) recover from this within the first two years.....most doctors say one year. But I find the number is closer to two years from talking to other preemie parents with the same problem. Most have babbling, wonderful little talkers. Most of the babies with this it's just their cry that can be affected, it's very weak and raspy sounding. Almost like they are out of breath.
Sometimes (like with our son and drastically) it can affect their swallow, as the vocal cords form sort of the last means of protection for the airway when swallowing. So it really depends on how the cord is paralyzed, open or closed. Closed is usually better but does present it's own issues, such as airway restriction and breathing issues when sick. Open (like our son's) also has it's issues because it leaves the airway unprotected, so when the baby swallows some can go into their lungs (this is called aspirating).
I would find out how the cord is paralyzed. I see from your avatar picture your son is likely off oxygen, so this is a great sign! Has he had any success at breast feeding or bottling? You may notice he destats or bradys more, or chokes when he drinks...if you do (and you may not at all) I would ask about having a feeding study done (especially if the vocal cord is paralyzed open). This will tell you and the doctors if he is aspirating and then they can thicken his food accordingly to make it safe for him to swallow. As too much aspirated liquid can cause lung issues and damage.
If you have any questions...feel free to PM me!
Good luck. Try to not worry, most kids recover from this just fine...and even if they don't like our son, they still have awesome and healthy futures ahead of them!
. Shane was extubated on his 15th day of life, and went straight to nasal cannula (but he did need some vapotherm for a while; he skipped CPAP entirely). I learned today from SweetPea's attending physician that the ENT does not want to "commit" yet to actual paralysis, the left vocal cord is however hypomotile. The right is compensating well, and I am glad that SweetPea (Shane)'s cry/voice is actually getting stronger. He is not aspirating, but since he has reflux issues I am still duly concerned. I'll make sure I have a copy of Shane's records so that I can get a copy of the laryngoscopy results. 
