The success of soft robots in displaying emergent behaviors is tightly linked to the compliant interaction with the environment. However, to exploit such phenomena, proprioceptive sensing methods which do not hinder their softness are needed. In this work we propose a new sensing approach for soft underwater slender structures based on embedded pressure sensors and use a learning-based pipeline to link the sensor readings to the shape of the soft structure. Using two different modeling techniques, we compare the pose reconstruction accuracy and identify the optimal approach. Using the proprioceptive sensing capabilities we show how this information can be used to assess the swimming performance over a number of metrics, namely swimming thrust, tip deflection, and the traveling wave index. We conclude by demonstrating the robustness of the embedded sensor on a free swimming soft robotic squid swimming at a maximum velocity of 9.5 cm/s, with the absolute tip deflection being predicted within an error less than 9% without the aid of external sensors.