Party dissension is something that most of us have had to deal with, either as a player or GM. How we handled it was sometimes not the best way. That could have stemmed from inexperience or youth. In any case dissension can lead to quick resolved arguments or players leaving.
As a player dissension is interesting because you are on one side of the argument. This is true even if you are neutral. That is still a side. You may be by yourself or have others with you. If you are alone and very stubborn the outcome may not be suitable to you. In most cases if the individual is alone they can be persuaded to come back into the fray. I remember a time during an adventure when a character refused to leave the town until he pick pocketed every citizen and plundered their homes. In the beginning it was fine and the GM did a couple of sidebars to keep him included in the adventure. Later it became annoying and we tried to pressure him into coming along. Nothing seemed to work until we threatened to kill his character. He finally relented and joined back up with us, but refused to do anything the rest of the night. He also refused to do anything for the next few adventures. He later left the group. That was not a very good way to deal with the dissension. I do not think that as a player much of anything would work with such an individual. The GM should step in these hard to handle situations.
Now if a few players have dissented then role playing may be the answer. Imagine being part of a group who is divided on a direction of travel or deciding the death of an NPC. Both scenarios are different but can cause the same angst. For the GM this can be an integral part of the adventure. Make the players role play the problem out. If they are role playing, feelings are less likely to be hurt. It is different when Gundar the barbarian is pressuring Smedly the thief to change his mind. Usually there is a decision made and the party continues on. A good GM will work this into the adventure and play on the characters’ arguments.
What happens when the party cannot agree? In the rare instances that I have experienced, we stopped play. As a GM I made sure that we discussed what happened and why. I took that information and made the next adventure better. On one occasion I was even able to pick up where the original adventure left off. I remember those problems as being caused by issues that were outside the realm of the characters and adventure. The players dissented because of issues they had with each other. During my playing and GM’ing years friends were never lost (true friends stick around) but players left. My hopes are that your groups are sound.