Characters do not know what a saving throw is.
Characters have no concept of game mechanics, although for things they are proficient in or have relevant knowledge of it may be assumed they know about the effects. Relevant
eg. A character does not know that you missed your saving throw against a fireball, what they know is they saw you fail to get out of the way and take the full force of a fireball to the face.
Witnessing a spell being cast
Xanthar's Guide to Everything expands upon the particular circumstances of witnessing a spell being cast.
Other spells, such as charm person, display no visible, audible, or otherwise perceptible sign of their effects, and could easily go unnoticed by someone unaffected by them. As noted in the Player’s Handbook, you normally don’t know that a spell has been cast unless the spell produces a noticeable effect. (Perceiving a caster at work. XGTE p85)
The command spell is a one-word command that the creature must follow if they fail the save. One such effect is
"Grovel. The target falls prone and then ends its turn"
You would witness a one-word command. Was it magic? Is the person issuing the command just hoping to be persuasive/intimidating?
To be perceptible, the casting of a spell must involve a verbal, somatic, or material component. (Perceiving a caster at work. XGTE p85)
RAW, a character that witnessed the command has just 'perceived a spell being cast'. However, XGTE continues to stipulate how one might determine the spell that was cast.
If the character perceived the casting, the spell’s effect, or both, the character can make an Intelligence (Arcana) check with the reaction or action. (Identifying a spell. XGTE p85)
It does not outline anything that allows the character to know anything about the success or failure of the spell. It seems that RAW one cannot know the results of a saving throw without seeing them directly (Or perhaps later investigation/divination in the case of something that might bestow a curse for example).
Knowing that your foe just cast Command on an ally and told them to Grovel may help guide your next action. Knowing any more than that would be up to DM fiat.
Failing a Death Saving Throw
There is nothing RAW that speaks of a noticeable, physical effect of failing or succeeding on a death saving throw. Therefore it remains a saving throw that your character is unaware of. Your DM could decide that you can see the unconscious creature getting worse/better, but again, this is DM fiat.
In my (very limited) experience, having the knowledge to help prevent the death of a friends PC is always better (removing player agency due to RAW and letting your friend's work go down the drain is not much fun for anyone at the table)