My take on an odd little point in the film-verse...
Bilbo was awoken from his pleasant dream (it involved a warm hearth, he remembered, a good foaming tankard and an old book of maps of his) by a foot in his side. Gentle but firm. Perhaps a bit more of the firm, dwarves being what they were. He briefly considered rolling over and trying to drift back into his dream as Bofur clambered back into his own bedroll besides him, then remembered whose company he was in. Thorin… well he could easily imagine what Thorin would say if he found he had skipped his watch. And, Bilbo thought miserably, most of it would be true.
Despite that, he still felt a spark of annoyance when he stood up to find Kíli pulling on his own cloak. "A double watch," he muttered, slightly bitter. Obviously they didn't trust him to stay awake on his own.
"Apparently I am too irresponsible to be left unchaperoned," Kíli murmured in response. The glint of humour in his eyes was enough to alleviate the supposed slight, though Bilbo still had some doubts.
The hobbit took a good look round to prove that he was doing his duty, then settled down to prod at the campfire's embers. "I thought you normally shared your watches with Fíli?"
"Alas, we are both considered too irresponsible to be left alone together."
Bilbo grinned, not particularly surprised. He knew that Thorin had 'had words' with the pair recently, but personally he didn't mind their sense of fun. Kíli paced around their camp, checking. He made a better job of it than Bilbo did, sword ready on his back, but still took the opportunity to leap nimbly over the recumbent Bombur's form.
"Is that what got you into trouble?"
"Yes." Kíli admitted. "I can manage a row of four. Bombur not included. Fíli can only manage three."
Bilbo glanced along the rows of snoring dwarves, and grimaced.
"Regrettably our uncle was last in line. He's well-hidden under that cloak." Kíli smiled a smile of happy reminiscence. "It was worth it though."
Briefly wondering whether he had unwittingly been involved in this competition, Bilbo watched as the young dwarf settled down next to his sleeping brother and moved on to another question. "Which of you two is the youngest?"
Bilbo nodded to himself. "I thought so."
Kíli looked up at him sharply. "Why?"
"Well, you know… the beard. Lack of it, rather."
Kíli's face held an expression somewhere between embarrassment, irritation and amusement, and Bilbo was suddenly worried that he had offended him. After all, the rest of the dwarves all took their beards very seriously, to the extent that Bilbo was beginning to feel odd just for not having one… But Kíli merely raised a hand to his face to touch the light stubble there and said simply "It grows slowly." Then he softly tangled his hands in his sleeping brother's hair instead.
Bilbo watched his careful movements and decided not to ask. Perhaps this was a brotherly dwarf thing?
"I used to have a beard." Kíli said suddenly.
"You did?" Bilbo paused, looking over to where Glóin was using his as a pillow.
"Not as large as that," Kíli admitted, fingers busy.
"And you shaved it off?"
"No." The tone of disparagement made Bilbo suspect that his guess about the dwarves and their beards was right.
"What happened then?"
"I woke one morning and it was gone." Kíli kept his head down in concentration. Bilbo now realised that he was braiding Fíli's hair. Lots of thin, fine braids.
The dwarf looked up, a slight smile on his face. "It happened about a week before we came to Hobbiton, while we were travelling together." He gave his brother a sharp, yet loving, glance.
"Ah." Understanding suddenly dawned for Bilbo.
"He still hasn't confessed." Kíli reached over to bend a nearby low-hanging branch slightly closer to him. "But I'm certain it was him." His eyes narrowed. "And he keeps complimenting me on how youthful I look without it. As do the others." He carefully began to knot each of the individual braids to the twigs of the tree branch.
"Ah. Ah. Are you sure that's a good idea?"
"Yes." Kíli grinned. "But remember," he slipped seamlessly into an impression of Thorin, "that I am an irresponsible young dwarf with no real understanding of the big wide world, who has been cruelly led astray by his elder brother."
"Of course." Bilbo sat next to him, trying not to laugh. "Here, I know a better knot for that."
After that, Bilbo was unsurprised to find that he was never asked to share a watch with either of the brothers again.