Yes, there are people whose sole job is "mission design." They design the trajectory that a spacecraft should follow to fulfill its mission and all of the maneuvers needed for that to happen. Some companies specialize in mission design consulting, like Space Exploration Engineering, Advanced Space LLC, or the X Team from JPL (although the latter are purely on a short-term consulting and advising role).
Typically, the mission designers will work with the initial funders of a mission to understand the end-goal of the mission (e.g. "fly over this part of the Earth every 3 days" or "orbit Mars such that we can map it with this precision within these latitude ranges"). From there, they'll work out what is the most appropriate final orbit the spacecraft should be in.
Then, they find the set of transfer orbits which lead to that final mission orbit (also called the station orbit). This involves placement of the maneuvers throughout the transfer trajectory. This leads to a "delta-v" budget, which the sum of velocity changes needed to reach the final orbit. This budget is important for the systems engineering team which will use it to scale the thrusters and select the propellant(s) for this mission. If it's a low-thrust mission, then the full design is done with that in mind (the control laws and trajectory optimization is quite different).
Once the thrusters are selected, the manufacturer will provide some thrust firing errors, e.g. "instead of being on for two minutes exactly, this thrust may be on for only 1min58 or up to 2min02." This information is communicated back to the mission designers who will run Monte Carlo analyses to determine the minimum and maximum fuel use to be expected. This may lead to a re-optimization of the trajectory.
There will be several iterations of this design process throughout the building of the mission as the destination orbits are changed (for objective alterations or for regulatory purposes enforced by the International Telecommunications Union).
Once the spacecraft is launched, the navigation team will try to accurately pinpoint of the position of the spacecraft after it has been deployed by the rocket. From that information ("launch errors"), the mission designers will recompute the time and thrust profiles of each burn in order to meet the mission objectives.