"My mind to your mind, my thoughts to your thoughts." From there, I don't think there was a lot of consistency in the line...sometimes, Spock and the other melder would recite their names, like "We are Spock, we are..."
Other times (like with the Horta from "Devil in the Dark," if I recall the scene and the ep name correctly), Spock simply gave voice to the other's thoughts, allowing Kirk, et al, to converse with the creature.
In other instances, I don't think there was any speech at all until the melders began to share thoughts and speak them aloud.
In Star Trek III, the scene with Sarek was entirely different. "Kirk, I must have your thoughts."
Once they began the mind meld, Sarek began to confirm things that he saw in Kirk's mind.
"You spoke of your friendship."
"Yes..."
"He told you not to grieve."
"Yes..."
Then, his voice began to take on his son's qualities.
"I haven't been, and always shall be, your friend."
"no...."
(cue Vulcan mystical music
)
In Trek VI, where Spock melded with Valeris to gain information, there was no "intro," so to speak. He simply entered her mind and began to search (the novelization had a nice description of what was going on within) for the information, and both participants recited the names. It was also evident that Valeris' fear and pain (?) as he searched for the location of the conference was shared by Spock (his voice cracked as he informed Kirk that "She does not know").
Of course, this is all just off the top of my head...I'd have to review the eps and movies in question to be certain.
Davy Jones
"Frightened? My dear, you are looking at a man who has laughed in the face of death, sneered at doom, and chuckled at catastrophe! I was petrified."
-- The Wizard of Oz