When merged, Shaman increases the rank of a random unit on your partner's side of the battlefield.
Your priority should be to merge Mime with Shaman, not Shaman with Shaman or Mime with Summoner for example.
Merging Summoner with another Summoner is fine. Summoner is here mostly to have Shaman available at all times and keep the merges happening.
Here are some of the most popular support teams:
1. The Demonologist is a legendary special unit that when merged, summons a monster to both players' fields, killing the monster gives mana to both players.
If you keep your board full you will have more and better merge options. Make sure you don't overwhelm your partner with too many spawned mobs, I recommend merging Demonologist at the end of the wave, when most or all monsters are dead. Merging Demonologist during a boss is generally not recommended.
2. Summoner is included to avoid a Shaman shortage, not to tier up your units. The higher tier your units are, the harder it is to merge them.
3. The Clock of Power is a legendary support unit which in Co-op raises the attack speed of another player's units of the same merge rank.
As of version 4.0 a new ability has been added. After merging in Co-op, it significantly increases the attack speed of the other player's units for 8 seconds.
4. Demonologist produces mana for both players, but mostly only you will be needing/using it.
Other support teams
Here are a few non-Shaman support teams I have seen out there:
1. Trapper periodically throws a net at a random path point. The net temporarily slows down monsters and also increases the damage they take. Summoner multiplies your units, allowing you to upgrade them to higher tiers. Mime helps you merge Summoner and Demonologist.
2. Demonologist provides mana to both players and Reaper has a chance to instantly kill a normal mob.
3. Trapper slows and debuffs the enemies. You will need to have multiple copies of him for later, tougher waves.
4. Hex deals with the normal mobs and Trapper slows/debuffs all enemies, but you need a teammate that can kill the bosses.