For data estimation and many other tasks in communication systems model-based methods have been employed for several decades. However, these methods also have some drawbacks. The optimal model-based methods are often extremely computationally expensive, while complexity reduced methods sometimes perform significantly worse than the optimal ones do. Further, available data cannot be incorporated into the estimation process. The incredible success of data-driven machine learning methods in many different application areas has recently led to investigations of the applicability of machine learning methods, mostly neural networks, for data estimation, but also for other tasks in communication engineering. These methods are expected to resolve the issues of model-based methods, but difficulties arise for the machine learning approaches as well. In this work, neural network based data estimators are investigated for a communication system utilizing the so-called unique word orthogonal frequency division multiplexing (UW-OFDM) transmission scheme through simulations. More precisely, in the simulations three different neural network architectures are employed for data estimation in two communication systems with different system dimensions. For both systems two modulation alphabets and two distinct ways of generating UW-OFDM symbols are considered, and both channel coded and uncoded transmission is regarded. The neural network based estimators are compared with model-based methods, whereby the achieved bit error ratios at specific signal-to-noise-ratio ranges serve as a performance measure. Further, the impact of data pre-processing on the estimation performance of the neural network based data estimators is investigated. Finally, the distribution of their estimates is examined and a brief complexity analysis of the neural networks is conducted.