In this study, we investigate two widely recognized interacting dark energy (IDE) models and assess their compatibility with observational data, focusing on the growth rate of matter perturbations. We explore IDE models with different equations of state (EoS) parameters for dark energy (DE), including the CPL parameterization and a constant value for $ w_{\mathrm{de}} $. To constrain the parameters of the IDE models using background data, we employ a Markov Chain Monte Carlo (MCMC) analysis. Our results show that both IDE-I and IDE-II models are Compatible with observational data, although with slight variations influenced by the homogeneity or clustering of DE. Following that, we investigate the growth of matter perturbations and perform a comprehensive statistical analysis utilizing both the background and growth rate data. The growth rate in IDE models exhibits deviations compared to the $\Lambda \mathrm{CDM}$ model due to the impact of homogeneity or clustering of DE, as well as the selection of the EoS parameter. However, we find that the IDE models show good compatibility with the growth rate data. Furthermore, we explore how the clustering or homogeneity of DE and the selection of the EoS parameter affect the evolution of the relative difference in the growth rate of IDE models, $\Delta f $, in comparison to the $\Lambda \mathrm{CDM}$ model. Lastly, we employ the AIC and BIC criteria to evaluate and identify the best model that is compatible with the observational data. The selection of the model depends on the homogeneity or clustering of DE, the EoS parameter, and the dataset used. Overall, the IDE-I and IDE-II models exhibit agreement with the data, with slight deviations depending on specific scenarios and parameters.
