Join size estimation on sensitive data poses a risk of privacy leakage. Local differential privacy (LDP) is a solution to preserve privacy while collecting sensitive data, but it introduces significant noise when dealing with sensitive join attributes that have large domains. Employing probabilistic structures such as sketches is a way to handle large domains, but it leads to hash-collision errors. To achieve accurate estimations, it is necessary to reduce both the noise error and hash-collision error. To tackle the noise error caused by protecting sensitive join values with large domains, we introduce a novel algorithm called LDPJoinSketch for sketch-based join size estimation under LDP. Additionally, to address the inherent hash-collision errors in sketches under LDP, we propose an enhanced method called LDPJoinSketch+. It utilizes a frequency-aware perturbation mechanism that effectively separates high-frequency and low-frequency items without compromising privacy. The proposed methods satisfy LDP, and the estimation error is bounded. Experimental results show that our method outperforms existing methods, effectively enhancing the accuracy of join size estimation under LDP.