The laser guide stars are indeed outside the science field of view. In the setup for the Wide Field Mode of the MUSE instrument, there is a mirror with a large hole in the middle: the light for the instrument goes through the hole, while the light from the adjacent part of the sky, where the laser guide stars appear (and also the natural star that's used as the "tip-tilt" reference in the Wide Field Mode) is reflected off to the side to the adaptive-optics system ("GALACSI").
Here's a figure (from the MUSE Instrument Description web page) showing the possible positions of laser-guide stars (LGS), and the tip-tilt star (TTS) for MUSE's Wide Field Mode (the TTS can be any reasonably bright star within the GALACSI field; it doesn't have to be in the upper right). The green square is the science field of view; the gray annulus is the part that gets reflected to the adaptive-optics system:
The alternate Narrow Field Mode allows for proper AO correction down to (in principle) the diffraction limit of the telescope. In this case, all the light goes through the central aperture, and then a special dichroic mirror reflects the laser wavelengths to part of GALACSI, while another dichroic reflects all the near-infrared light to a different part of GALACSI, where the TTS star's light (here called the on-axis guide star = OGS) is analyzed; the parts of the optical light not reflected by the sodium-wavelengths dichroic go to the actual science instrument.
(See here for more details.)
But, yes, scattered light from the laser guide stars is a concern. I believe that the Narrow Field Mode deals with thus using the aforementioned sodium-wavelength dichroic; for the Wide Field Mode, there is a narrow-band filter in the science instrument to block light from the laser. Both approaches do exclude light from astronomical sources with the same wavelengths, so you have to plan your observing accordingly. (E.g., if you really want to observe those wavelengths, then you can't use the laser guide stars.)
The figure below (from the MUSE user manual) shows the instrument throughput in the Wide Field Mode as a function of wavelength. The blue and red curves show sub-modes using the laser guide stars ("AO" = adaptive optics), and the fact that they go to zero for wavelengths of $\sim 5700-6000$ Angstroms is due to the filter being used for those modes.