Locally differentially private (LDP) graph analysis allows private analysis on a graph that is distributed across multiple users. However, such computations are vulnerable to data poisoning attacks where an adversary can skew the results by submitting malformed data. In this paper, we formally study the impact of poisoning attacks for graph degree estimation protocols under LDP. We make two key technical contributions. First, we observe LDP makes a protocol more vulnerable to poisoning -- the impact of poisoning is worse when the adversary can directly poison their (noisy) responses, rather than their input data. Second, we observe that graph data is naturally redundant -- every edge is shared between two users. Leveraging this data redundancy, we design robust degree estimation protocols under LDP that can significantly reduce the impact of data poisoning and compute degree estimates with high accuracy. We evaluate our proposed robust degree estimation protocols under poisoning attacks on real-world datasets to demonstrate their efficacy in practice.