In tiny independent teams, there can often be no space for designated product people (a term used to describe producers, game designers, and live operations managers). In such situations, game design responsibilities are distributed between the artists and programmers.
However, while a team with no designer may work well on a small indie scale (one to four people), the prospect of running a game team of more than five people without a designated designer is a very risky one. With the freedom of distributed design (colloquially called design by committee) often comes a lack of ownership and accountability. Making good design decision becomes increasingly difficult as the game is being pulled in different directions and lacks a cohesive vision. In such teams, even the most pressing gameplay issues can go unaddressed. Everyone loves to chip in with their ideas, but nobody feels compelled to work on design problems, be it for the fear of associating themselves with them, or the conflict of going against the rest of the team.
You're most likely going to be a part of a small or mid-sized team. Anything with 15 to 30 people would be considered a medium-sized team, and this number will always fluctuate depending on where you are in the development cycle and includes a total of two to four design-oriented staff members. A smaller operation has a headcount of 5 to 12 people, and one or two other design-oriented people at most. It's also quite common, especially for a senior designer working on games with a relatively manageable scope, to be the only designated game designer on the project.
Each company has its own approach to managing human resources. And while various artists and even programmers jump in and out of projects to help the teams around critical milestones, designers are often in it for the long run, that is, from preproduction until the game is released, and most likely some more after that. This is not only because of the importance of having a unified design direction but also due to the sheer amount of time and effort required to get to know the game and make meaningful design decisions.
While bigger teams have hugely varying structures, with space for a very low-level specialization, medium and small teams usually share a similar setup across the industry. As a great example of the strategy of small teams, let's look at the creators of Clash of Clans, the Finnish game company Supercell, who by definition are an assembly of independent cells. Each cell is a small team consisting of a producer, a few artists and coders, at least one tester, a generalist game designer and potentially a monetization/live operations-oriented person. People rotate in and out of their cells slowly as games are kicked off or killed (canceled). Each team can also count on support from a stable of people who work for the whole company, taking care of player analytics, finance, social media, operations, customer support, marketing, user acquisition, and more. The goal of Supercell's structure is to create an environment that facilitates creativity, and gives each team the power to make decisions on a game's design and direction.