A variety of deep neural network (DNN)-based image denoising methods have been proposed for use with medical images. These methods are typically trained by minimizing loss functions that quantify a distance between the denoised image, or a transformed version of it, and the defined target image (e.g., a noise-free or low-noise image). They have demonstrated high performance in terms of traditional image quality metrics such as root mean square error (RMSE), structural similarity index measure (SSIM), or peak signal-to-noise ratio (PSNR). However, it has been reported recently that such denoising methods may not always improve objective measures of image quality. In this work, a task-informed DNN-based image denoising method was established and systematically evaluated. A transfer learning approach was employed, in which the DNN is first pre-trained by use of a conventional (non-task-informed) loss function and subsequently fine-tuned by use of the hybrid loss that includes a task-component. The task-component was designed to measure the performance of a numerical observer (NO) on a signal detection task. The impact of network depth and constraining the fine-tuning to specific layers of the DNN was explored. The task-informed training method was investigated in a stylized low-dose X-ray computed tomography (CT) denoising study for which binary signal detection tasks under signal-known-statistically (SKS) with background-known-statistically (BKS) conditions were considered. The impact of changing the specified task at inference time to be different from that employed for model training, a phenomenon we refer to as "task-shift", was also investigated. The presented results indicate that the task-informed training method can improve observer performance while providing control over the trade off between traditional and task-based measures of image quality.