In multi-agent planning, preserving the agents' privacy has become an increasingly popular research topic. For preserving the agents' privacy, agents jointly compute a plan that achieves mutual goals by keeping certain information private to the individual agents. Unfortunately, this can severely restrict the accuracy of the heuristic functions used while searching for solutions. It has been recently shown that, for centralized planning, blind search algorithms such as width-based search can solve instances of many existing domains in low polynomial time when they feature atomic goals. Moreover, the performance of goal-oriented search can be improved by combining it with width-based search. In this paper, we investigate the usage of width-based search in the context of (decentralised) collaborative multi-agent privacy-preserving planning, addressing the challenges related to the agents' privacy and performance. In particular, we show that width-based search is a very effective approach over several benchmark domains, even when the search is driven by heuristics that roughly estimate the distance from goal states, computed without using the private information of other involved agents. Moreover, we show that the use of width-based techniques can significantly reduce the number of messages transmitted among the agents, better preserving their privacy and improving their performance. An experimental study presented in the paper analyses the effectiveness of our techniques, and compares them with the state-of-the-art of collaborative multi-agent planning.