How about consensus?
1) An opinion or position reached by a group as a whole
2) general agreement or accord
A consensus refers to a collective or group opinion, which seems to be what you're looking for - that the decision in question is often held by individuals, or else that people generally agree to the opinion once articulated.
It hopefully doesn't have the same baggage as "democratic" - I'm actually not sure what baggage you're running across with the word, but I'm sort of guessing the problem is something like democracy as "electing a single authority to have the power or make the decision", not democracy as "propose possibilities then everybody directly vote on the alternative solutions".
Uses might include "government by consensus", or "reached an agreement through consensus", or "used a consensus system to make the decision fair".
Or else perhaps concurrence
- Agreement in opinion.
- Cooperation, as of agents, circumstances, or events.
- Simultaneous occurrence; coincidence.
"They reached an agreement by concurrence" or "their system of concurrence means their decisions are mutually acceptable" or something like that. It's a bit less easily used than consensus, in my opinion, takes a bit more thinking to fit into sentences, but it may work for what you need it to be.
You might also like accord, though it's a bit clunkier
- agreement; conformity; accordance (esp in the phrase in accord with)
- consent or concurrence of opinion
- with one accord unanimously
- a settlement of differences, as between nations; compromise
- to be or cause to be in harmony or agreement
This would be used like "they reached an accord" to make a decision, or "they were in accord with each other" about their system.