Online Decentralized Learning (ODL) is suitable for Internet-of-Things (IoT) devices since only parameter updates are exchanged with neighbors to avoid uploading private data to a central server and the training data is allowed to arrive at the devices sequentially. However, the current ODL frameworks cannot support the emerging Social IoT (SIoT) paradigm favorably since the SIoT devices exchange parameter updates with only trust-worthy neighbors based on specific social relations (e.g., parental object relation and ownership object relation). Conversely, sharing parameter updates with untrustworthy neighbors could speed up the training process but may violate social relations. Differential privacy (DP) is thus used to ensure data security while excessive devices engaging DP may downgrade the training performance. However, most research neglects the effect of neighbor selection for each device based on social networks, physical networks, and DP. Thus, in this paper, we innovate an ODL framework ODLF-PDP to allow only a part of devices to engage DP (i.e., partially DP) to improve training performance. Then, an algorithm BeTTa is proposed to build an adequate communication topology based on the interplay among the social networks, physical networks, and DP. Last, the experiment results manifest that ODLF-PDP saves more than 20% physical training time compared to the current frameworks via the benchmark of MNIST.