White noise has distinct statistical properties that allow to mitigate it's effect on detection of meaningful signal. E.g. filtering white noise from audio stream is a very common operation.
Sending packets to all contacts at random is a form of introducing white noise, vulnerable to signal processing techniques known and used from 1950s.
Your noise can be non-white. Your noise can favour some of your peers, some time of day, messages can be elaborately routed around in circles. You can even make clients download new message distribution patterns each day. Genetically enchanced patterns.
If your noise is algorithmic, and even worse, the algorithm is known (as would be the case of an open source project), it makes your protection fairly vulnerable.
E.g. in case of biasing the method towards time of day, the attacker likely can filter it out using fairly basic statistical methods. Generally any kind of pseudo-randomization would render the method vulnerable; at least to the caliber of mathematicians which work at NSA.
If there's messaging system's own statistical analyzer, that learns your communication patterns and adjusts false messaging to rectify that, it could work. However the other messaging clients have to play along, which makes it fairly challenging problem.
Sending packets to all contacts at random is a form of introducing white noise, vulnerable to signal processing techniques known and used from 1950s.