An important task in functional data analysis is to estimate a smooth function based on sparse and noisy observations on a time interval. A specific form of sparsity may be introduced during data collection due to occlusions or dropouts where an entire segment of a function is missing. To address this problem, we define a Bayesian model that can fit individual functions on a per subject basis, as well as multiple functions simultaneously by borrowing information across subjects. A distinguishing property of this work is that our model considers amplitude and phase variabilities separately which describe y-axis and x-axis variability, respectively. For this purpose, we use the square-root velocity representation of functions. For the multiple function estimation problem, we assume a common low-dimensional template function to model amplitude and use subject-specific warping functions to model phase. This setup allows these fitted functions to share common features while allowing uncertainty quantification in terms of the two components separately. We validate the proposed framework using multiple simulated examples as well as real data including ECG signals and growth curves.