Laser speech detection uses a non-contact Laser Doppler Vibrometry (LDV)-based acoustic sensor to obtain speech signals by precisely measuring voice-generated surface vibrations. Over long distances, however, the detected signal is very weak and full of speckle noise. To enhance the quality and intelligibility of the detected signal, we designed a two-sided Linear Prediction Coding (LPC)-based locator and interpolator to detect and replace speckle noise. We first studied the characteristics of speckle noise in detected signals and developed a binary-state statistical model for speckle noise generation. A two-sided LPC-based locator was then designed to locate the polluted samples, composed of an inverse decorrelator, nonlinear filter and threshold estimator. This greatly improves the detectability of speckle noise and avoids false/missed detection by improving the noise-to-signal-ratio (NSR). Finally, samples from both sides of the speckle noise were used to estimate the parameters of the interpolator and to code samples for replacing the polluted samples. Real-world speckle noise removal experiments and simulation-based comparative experiments were conducted and the results show that the proposed method is better able to locate speckle noise in laser detected speech and highly effective at replacing it.