The Monster Manual's Statistics chapter (p6) opens as such:
A monster's statistics, sometimes referred to as its stat block, provide the essential information that you need to run the monster.
The crucially, the Druid's Wild Shape states:
Your game statistics are replaced by the statistics of the beast, but you retain your alignment, personality, and Intelligence, Wisdom, and Charisma scores. You also retain all of your skill and saving throw proficiencies, in addition to gaining those of the creature. If the creature has the same proficiency as you and the bonus in its stat block is higher than yours, use the creature's bonus instead of yours. If the creature has any legendary or lair actions, you can't use them.
If only temporarily, the Druid's statistics are replaced by the beast's statistics and during that timeframe the player needs to be able to use those statistics to do anything at all.
This is a case of specific beats general:
- In general, the Monster Manual is a tool exclusive to the DM.
- In the specific case where a player needs to use the statisitcs of a monster, the Statistics chapter applies to the player as well.
If the Statistics chapter does not apply to the player, then then the player is incable of using all the things explained within that chapter which include hit points, speeds, senses, attacks, and so on. That is absurd.
Given that the Statistics chapter must apply to a player using monster statistics, then so does the Melee and Ranged Attacks substection (p10). Hence, when a player hits with a monster's attack they choose, as explained in the subsection:
Hit. Any damage dealt or other effects that occur as a result of an attack hitting a target are described after the "Hit" notation. You have the option of taking average damage or rolling the damage; for this reason, both the average damage and the die expression are presented.
For example, the statistics of a beast such as a Brown Bear state:
Bite. Melee Weapon Attack: +6 to hit, reach 5 ft., one target. Hit: 8 (1d8 + 4) piercing damage.
Whoever runs the bear gets to choose between 8 or 1d8+4 damage. In general, that's the DM. In the specific case of a Wild Shaped Druid, that's the player.