The immune system is one of the first lines of defence against the emergence of cancer. When effector cells attempt to suppress the tumour, the cancer cells can respond in kind by evolving methods of escape or inhibition. Knowledge of this coevolutionary system and the selection taking place within it can help us understand tumour-immune dynamics both during tumorigenesis but also when treatments such as immunotherapies are applied. Here, we present an individual-based branching process model of mutation accumulation, where random mutations arising in cancer cells trigger corresponding specialised immune responses. Different from previous research, we explicitly model interactions between cancer and effector cells, while incorporating stochastic effects, which are especially important for the expansion and extinction of small populations. We find that the parameters governing interactions between the cancer and effector cells induce different outcomes of tumour progress, such as suppression and evasion. While it is hard to measure the cancer-immune dynamics directly in patients, genetic information of the cancer may indicate the presence of such interactions. Our model demonstrates signatures of selection in sequencing-derived summary statistics, such as the single-cell mutational burden. Thus, bulk and single-cell sequencing of a tumour may give information about the coevolutionary dynamics.