Rules as Intended: the DM chooses the monster
The rules as written does not define who gets to pick what or how. However, the Sage Advice Compendium clarifies how the spell was intended to work by the designers:
Some spells of this sort specify that the spellcaster chooses the creature conjured. For example, Find Familiar gives the caster a list of animals to choose from. Other spells of this sort let the spellcaster choose from among several broad options. For example, Conjure Minor Elementals offers four options. The design intent for options like these is that the spellcaster chooses one of them, and then the DM decides what creatures appear that fit the chosen option.
So it is clearly intended that the DM is able to pick the monster after the player picks which option they want.
There is no indication of how the DM would choose the beast, so that is left entirely up to the DM at your particular table to decide by the designers.
The player can only choose the broad option, not the CR or creature
For conjure animals, the spellcaster has 4 options:
- One beast of challenge rating 2 or lower
- Two beasts of challenge rating 1 or lower
- Four beasts of challenge rating 1/2 or lower
- Eight beasts of challenge rating 1/4 or lower
Again, according to RAW, there is no indication who picks what. But, the SAC as quoted above does give a hint as to the intent:
[...] Other spells of this sort let the spellcaster choose from among several broad options. For example, Conjure Minor Elementals offers four options. The design intent for options like these is that the spellcaster chooses one of them [...]
The options the spellcaster chooses from are broad. In other words, the implication is that they don't have any control over the fine details of the option they pick. Thus, the spellcaster, under this RAI interpretation, would only be able to pick one of the 4 options and everything else (CR and specific beast) would be up to the DM.
Of course, there is no reason a DM would have to run the spell this way, this just seems to be the way the designers intended. In fact, as a DM I often have the player pick the creature with an option to veto the choice.
If you are planning on taking and/or using the spell ask your DM ahead of time what method they will take and how you can help make it easy on them.