How do hand-seals work when performed with another participant? This is related to: What is the difference when a jutsu is performed by weaving only some of it's seals and when performed by weaving all of them?

According to the answer, head-weaving is a technique used to manipulate chakra to perform the Jutsu. Some users require more hand seals to perform the Jutsu whereas other do not require as many due to their chakra control. 

In Naruto Shippuden, Episode 374


   We see Kakashi and Obito having their battle in the realm where their Mangekyou Sharingan takes them. At roughly 8:30, Kakashi and Obito are in close quarters and Obito takes Kakashi's hand to perform the hand-seals for FireBall Justu.


How does the charka manipulation through hand-seals work when performing them with another participant?
As said in the linked post, the hand signs are for the manifesting chakra, while weaving hand signs. More the shinobi's skill and chakra control increases, less the hand signs he needs to use. While techniques may require a number of hand seals to work, a skilled ninja can use less or even one to perform the same technique.

In the whole series, we've seen shinobi using hand signs with two hands, one hand and even no hand signs, which requires a very precise control of chakra. As mentioned in this post, at various instances shinobi of great skill like Itachi, Minato, Sasuke or Naruto used even one hand signs or no hand signs.

But When using hand signs with another participant, they both need their chakra to be in perfect sync with each other and should have a great control over it. That's why Naruto and Sasuke could accomplish some jutsu. 

Also, there were instances, like in  Obito Uchiha using Kakashi Hatake's hand for the Fire Release: Great Fireball Technique or  Anko Mitarashi using Orochimaru's hand for the Twin Snakes Mutual Death Technique. For these instances, a shinobi needs to have a great amount of chakra, which he forces to make it flow into other participant's body and gains control of other's chakra (momentarily) to get his jutsu done.

Hope this answers your doubt. :)