Both my DMs say that if you have the components for the spell you can cast the spell without using a spell slot. Is this true normally in D&D 5e, or is it more of a DM discretion thing?
What your DMs are describing is NOT standard 5e. I don't know where they're getting their information from, but it has to be some kind of (clearly misrepresented) homebrew mechanics.
It's possible they've confused things with the Ritual Mechanics:
Certain spells have a special tag: ritual. Such a spell can be cast following the normal rules for spellcasting, or the spell can be cast as a ritual. The ritual version of a spell takes 10 minutes longer to cast than normal.
It also doesn't expend a spell slot, which means the ritual version of a spell can't be cast at a higher level.
To cast a spell as a ritual, a spellcaster must have a feature that grants the ability to do so. The cleric and the druid, for example, have such a feature. The caster must also have the spell prepared or on his or her list of spells known, unless the character's ritual feature specifies otherwise, as the wizard's does.
Ritual spellcasting does allow a spellcaster to cast a spell without consuming a spell slot.
But that's completely different from the material requirements: if a spell specifies material requirements, you're required to provide those components (which, if the spell specifically says they're consumed, they will be consumed), OR, you're allowed to use an Arcane Focus/Holy Symbol if (and only if) the material components do not specify a cost. Whether you provide the raw components or use an arcane focus has no bearing on whether a spell slot is consumed; only cantrips and spells cast as rituals preserve spell slots (or racial/class features that allow specific spells to be cast without using spell slots).
Neither, they don't seem to know how spellcasting works in 5e
All spells have required components that need to be provided for them to be cast. Verbal, somatic, and material are all possibilities.
In order to cast any spell, the caster must provide all the components required unless a specific condition negates the need (i.e. Subtle Spell Metamagic eliminates verbal and somatic requirements). Focusing on the Material components, the PHB states the following:
Casting some spells requires particular objects, specified in parenthesis of the component entry. A character can use a component pouch or a spellcasting focus in place of the components specified for a spell.
Jumping over to the Druid (or really any caster):
To cast one of these druid spells, you must expend a slot of the spell's level or higher.
This is true for any caster. If you have no slots left at the time of casting, then you cannot cast the spell. If you cannot provide all of the verbal, somatic, and material components for a spell, then you cannot cast the spell.
The only means to cast a spell without using a spell slot is to cast the spell ritually and this can only be done by characters whom have the Ritual Casting feature, have a spell with the ritual tag prepared in some manner to ritually cast, and can spend the additional time required to complete the ritual.
Your DM's are misunderstanding spell casting mechanics. Spell casting is covered in chapter 10 of the Player's Handbook (pp 202-207). Casters prepare spells that they can cast, then use spell slots to cast them. Cantrips are an exception. They do not use spell slots and can be cast at will provided you can meet the components requirement.
In short by RAW (rules as written):
When a character casts a spell, he or she expends a slot of that spell's level or higher.
A spell's components are the physical requirements you must meet in order to cast it...If you can't provide one or more of a spell's components, you are unable to cast the spell.
MATERIAL(M) Casting some spells requires particular objects,specified in parentheses in the component entry. A character can use a component pouch or a speIlcasting focus (found in chapter 5) in place of the [material] components specified for a spell. But if a cost is indicated for a component, a character must have that specific component before he or she can cast the spell.
There is one way to cast some spells without using a spell slot:
Certain spells have a special tag: ritual. Such a spell can be cast following the normal rules for spell casting, or the spell can be cast as a ritual. The ritual version of a spell takes 10 minutes longer to cast than normal. It also doesn't expend a spell slot, which means the ritual version of a spell can't be cast at a higher level. To cast a spell as a ritual, a spellcaster must have a feature that grants the ability to do so.