Tensegrities form a special case of truss, wherein compression members (struts/bars) float within a network of tension members (cables). Tensegrities are characterized by the presence of at least one infinitesimal mechanism stabilized with member pre-stress to ensure equilibrium. Over prolonged usage, the cables may lose their pre-stress while the bars may buckle, get damaged, or corrode, affecting the structural stiffness leading to change in the measured dynamic properties. Upon loading, a tensegrity structure may change its form through altering its member pre-stress affecting its global stiffness, even in the absence of damage. This can potentially mask the effect of damage leading to a false impression of tensegrity health. This poses the major challenge in tensegrity health monitoring especially when the load is stochastic and unknown. Present study proposes an output-only time-domain method that makes use of tensegrity vibrational responses within a Bayesian filtering-based approach to monitor the tensegrity health in the presence of uncertainties due to ambient force, model inaccuracy, and measurement noise. For this, an interacting strategy combining Particle Filter (PF) and Ensemble Kalman Filter (EnKF) has been adopted (Interacting particle-Ensemble Kalman Filter, IP-EnKF) in which the EnKF estimates the response states as ensembles while running within a PF envelop that estimates a set of location-based health parameters as particles. Furthermore, for a cheaper damage detection procedure, strain responses are used as measurements. The efficiency of the proposed methodology in terms of accuracy, computational cost, and robustness against noise contamination has been demonstrated using numerical experiments performed on two tensegrity modules: a simplex tensegrity and an extended-octahedron tensegrity.