Anyone can attempt to do that
Anyone can attempt to do anything, whether they have the feat to do it or not (even try to recall everything accurately, as in Keen Mind). It doesn't mean they will always succeed, but it also doesn't mean they need the feat to make an attempt.
This is an example of bad wording, for a feat
The way this feat is framed implies that only characters with this feat can succeed. Drawing on the wisdom from the Unearthed Arcana: Feats, specifically on the "bad feat" (as identified in the article) named "Warhammer Master," we learn the following:
The ability to knock aside an opponent's shield is nifty -- but that's something any character should be able to attempt. Locking that down into a feat threatens to limit the game's flexibility. You could argue that anyone could still try that trick, but the way the feat frames the ability makes it sound like only characters with this feat can succeed. This option is an area that I'd want DMs to adjudicate on their own, rather than bloating the game with fiddly rules.
This is in response to this part of the Warhammer Master feat:
In addition, you can use your warhammer to knock away a foe's shield. If you hit a creature that’s using a shield, you can have the attack deal no damage and force the creature to drop its shield.
The design goal, as far as Mike Mearls was concerned, was to write feats in such a way that they did not limit the game's flexibility. The way this part of the Actor feat is written implies you need to have the Actor feat to even attempt to mimic someone's speech.
How do non Actors do it without stepping on the feat?
I believe the non-obvious but valid intent, which will still grant this ability to everyone else without necessarily needing to step on this feat's toes, was to imply that the Actor is so good, they can mimic anyone after one minute's worth of study. Everyone else has to take at least a few hours before they can even try properly.
Listen to a chirping bird for a minute, or a bubbling brook, and try to mimic that sound. You might be able to, but you probably can't; but, someone in the world probably could (out of the 7B people on Earth, chances are pretty high, right?)
This feat makes your character so good, they will find that person is themselves, for any sound or speech pattern.
What do they roll?
The Actor feat gives us the guidance for what skill to use:
A successful Wisdom (Insight) check contested by your Charisma (Deception) check allows a listener to determine that the effect is faked.
This makes sense as you are trying to deceive someone that you are what you are not. Based on this feat, then in most cases, you should roll Deception vs Insight. However, for non-Actors, you could use different skills as you see fit.
A Charisma (Performance) check could replace Deception, if you interpret this particular check as a test of how masterfully they can mimic another sound. Perhaps they are openly mimicking robins singing to an audience of children.
A Charisma (Stealth) check could replace Deception, if they are trying to hide by creating noise that makes them blend into the environment.
If interpreted this way, then for the Actors, the predictability of what skill to use means they know where to put their proficiency and Expertise, should they acquire them. This makes this feat even more useful, because for Actors, they always know which skill to use.