[Malleys]

var data = {
	"user 1": {
		company: 'Рога и копыта',
		name: 'Александр',
		projects: {
			projectID1: {
				role_right: 0
			}
		}
	},
	"user 2": {
		company: 'Копыта и рога',
		name: 'Петя',
		projects: {
			projectID1: {
				role_right: 0
			}
		}
	},
	"user 3": {
		company: 'Копыта и рога',
		name: 'Василий',
		projects: {
			projectID1: {
				role_right: 2
			}
		}
	},
};

var temp = Object.values(data).reduce((object, user) => {
	if(user.company in object === false) {
		object[user.company] = {
			title: user.company,
			data: []
		};
	}

	object[user.company].data.push(user);

	return object;
}, {});

var newData = Object.values(temp).map(company => {
	company.data.sort((a, b) => a.projects.projectID1.role_right > b.projects.projectID1.role_right ? 1 : -1);

	company.data = company.data.map(user => {
		user.role_right = user.projects.projectID1.role_right;
		delete user.projects; // возможно не надо удалить?
		return user;
	});
	return company;
});

console.log(newData);