In the published adventure I run I think I've seen things along the lines of "a character proficient with lock picking tools may attempt to unlock this door with a DC15." So some tools (typically) require proficiency to even try (but it's written at the same place as the DC, not under the description of the tool).
No, adventures are badly written and go against the rules all the time, usually to the detriment of the game. I would heartily suggest that you parse out anything approaching a rule that appears in an adventure and apply what the rules actually are instead.
That means:
decide whether or not a roll is possible/needed
decide what stat is needed
decide the DC
ask for a check
decide whether players get to add proficiency bonuses if they ask
Other, such as cook's utensils, cartography, games etc make sense that anyone can try to use. Even smith's tools might be usable for some tasks without proficiency.
Actually someone pointed out to me that there aren't proficency checks in this edition. There are just ability checks, and proficiencies give a bonus to them. Proficiency with a tool allows you to add your proficiency to ability checks that use the tool, it doesn't allow you to use a tool.
Since we seem to have gotten mired in lockpicking, bear in mind that:
1. Medieval locks tend to be "if you can make an object that can reach this point through the keyhole, the lock is opened". The most complex might actually require more than 2 bent nails to achieve this.
2. Most modern locks, despite being much more advanced are trivially pickable by someone with a vague idea of what to do and enough time. I wouldn't consider myself dextrous, and I've definitely not spent 200 days training with lockpicks, and I've successfully picked a modern lock with a couple of home made tools. ie - bent bits of stiff wire. That puts the DC for a typical modern lock at ~20, possible with no bonus, but will take a while.
So in short: I would say that trying to pick a lock with zero tools is impossible. However the tools required are pretty minimal, and certainly something improvised should suffice for most locks. Finally, the level of expertise to pick all but the most advanced and expensive locks is pretty low.
Where an expert lock picker comes in is the speed with which he can pick a lock and the stealth with which he can do it.
In short - an unattended lock should be no more than a minor speedbump even to a party with no dedicated lock picker. The dedicated lock picker shines when you pair a locked door with a dangerous guard creature or timed event.