Aims. We present the updated census and statistics of Lyman-α emitting long gamma-ray bursts host galaxies (LAE-LGRBs). We investigate the properties of a sub-sample of LAE-LGRBs and test the shell model commonly used to fit Lyman-α (Lyα) emission line spectra. Methods. To perform the census we use all the LGRB host galaxies with relevant information presented in the literature or publicly available. Among the LAE-LGRBs detected to date, we select a golden sample of four LAE-LGRBs (GRBs: 011211, 021004, 060926, 070110) allowing us to retrieve information on the host galaxy properties and of its interstellar medium gas, through the combination of the analysis of their afterglow and host galaxy observations. We fit their Lyα spectra using the shell model, and constrain its parameters with the observed values.Results. The census results in 29 LAE-LGRBs detected to date. Among them, we present five new Lyα emission detections in hostgalaxy spectra and the corresponding unpublished VLT/X-shooter data (GRBs: 060926, 070110, 081121, 081222 and 100424A). From the comparison of the statistics and properties of LAE-LGRBs to those of LAE samples in the literature, we find evidences of Lyα suppression in dusty systems, and a fraction of LAE-LGRBs among the overall LGRB hosts lower than that found for Lymanbreak galaxy (LBG) samples at similar redshift range. This result can be explained by the different selection criteria of the parent samples and by the shallower spectral observations of LGRB samples compared to LBG ones. However, we find that LAE-LGRBs are representative of Lyα emission from the bulk of UV-selected galaxies at z ∼ 2. We find that the golden sample of LAE-LGRBs studied here are complex systems characterized by multiple emission blobs and by signs of possible galaxy interactions. The fitting procedure recovers the HI column densities (N HI ) measured from the afterglow spectra, and the other properties described by the shell-model parameters in the two low-N HI cases, but it fails in doing so in the other two with high N HI . The afterglows of mostLGRBs and LAE-LGRBs show log(N HI /cm −2 ) > 20.3, implying that statistically the bulk of Lyα photons expected to be produced by massive stars in the star-forming region hosting the GRB will be surrounded by such opaque lines of sight. We therefore interpret our results in the context of more sophisticated models and of different dominant Lyα emitting regions. We also compare LAE-LGRBs to LAE Lyman continuum (LyC) leakers in the literature in terms of properties identified as possible indirect indicators of LyC leakage. We find that only one LGRB (GRB 021004) would be a likely strong LyC leaker and discuss the validity of such indicators at high redshift.