The answer to your question depends on whether you are referring modes using all the same notes or modes using the same tonic.
To explain:
With the C Major scale,
C Major Scale:
CDEFGABC
the relative natural minor (aeolian mode) is
A Minor Scale:
ABCDEFGA
This uses only notes found in C Major, but now A is the tonic.
The parallel natural minor to C Major is
C Minor Scale:
C D EbF G AbBbC
This still has C as the tonic but changes the other notes to make it a natural minor scale instead of a major scale.
Likewise, C Major (C ionian) has what you might call "relative modes,"
Relative Dorian Scale:
D E F G A B C D
Relative Phrygian Scale:
E F G A B C D E
Relative Lydian Scale:
F G A B C D E F
Relative Mixolydian Scale:
G A B C D E F G
Relative Aeolian Scale:
A B C D E F G A
Relative Locrian Scale:
B C D E F G A B
and "parallel modes."
Parallel Dorian Scale:
C D EbF G A BbC
Parallel Phrygian Scale:
C DbEbF G AbBbC
Parallel Lydian Scale:
C D E F#G A B C
Parallel Mixolydian Scale:
C D E F G A BbC
Parallel Aeolian Scale:
C D EbF G AbBbC
Parallel Locrian Scale:
C DbEbF GbAbBbC
The chords that fall "naturally" in the scale are those which use only the notes of the scale; therefore, modes "relative" to C major will use the same chords, and modes "parallel" to C major will use different chords. (At least three chords will be different from mode to mode.)
Even in the "relative" modes to C Major, the roman numerals for each chord will change to reflect their different functions relative to the different tonics.
Ionian (Major):
C Dm Em F G Am B* (* means diminished, + means augmented)
I ii iii IV V vi vii*
Dorian:
DmEm F G AmB* C
i ii III IV v vi* VII
or usually
i ii bIII IV v vi* bVII (to indicate the position of the roots of each chord relative to the "parallel" major)
Phrygian:
EmF G Am B* C Dm
i II III iv v* VI vii
or usually
i bII bIII iv v* bVI bvii
Lydian:
F G Am B* C Dm Em
I II iii iv* V vi vii
or usually
I II iii #iv* V vi vii
Mixolydian:
G Am B* C DmEm F
I ii iii* IV v vi VII
or usually
I ii iii* IV v vi bVII
Aeolian (Natural Minor):
AmB* C Dm EmF G
i ii* III iv v VI VII
or sometimes
i ii* bIII iv v bVI bVII
Locrian:
B* C Dm Em F G Am
i* II iii iv V VI vii
or usually
i* bII bii iv bV bVI bvii
A Source: https://en.wikipedia.org/wiki/Roman_numeral_analysis#Modes
While the chords that fall "naturally" in the scale are the same in "relative" modes, it may be common to alter some notes in particular chords; for example, in minor keys it is very common to raise the 7th note of the scale (G-G#) in the V chord and the vii* chord, but not usually in the III chord (which would become a III+ chord). This is so common (at least in "common practice" classical music) that the major V chord is considered standard in minor keys even though doesn't fall "naturally" in the scale, and the minor v is actually considered abnormal (used to make a "modal" sound I'm told) even though it does fall in the scale "naturally." Other answers refer to alterations common in other modes, but I don't really know what the common alterations in other modes are.
The Wikipedia article on "Aeolian Mode" implies that people use the term "aeolian" only for music that avoids using harmonic and melodic minor, (which are frequently used in "ordinary" minor tonality,) and the article on mode says,
Indeed, when 19th-century composers revived the modes, they rendered them more strictly than Renaissance composers had, to make their qualities distinct from the prevailing major-minor system. Renaissance composers routinely sharped leading tones at cadences and lowered the fourth in the Lydian mode (Carver 2005, 74n4).
https://en.wikipedia.org/wiki/Mode_(music)#Use_2
Also, it occurs to me that you may be asking if the most common chord progressions are different in different modes. As you mentioned, this chord progression
"50s Progression:"
I vi IV V <repeat>
C Am F G
is very common in popular music. This would sound completely different in phrygian,
"50s Progression" in Phrygian
i bVI iv v* <repeat>
Em C Am B*
so maybe some other progression like this:
My Made-up Phrygian Chord Progression:
i bIII iv bII <repeat>
Em G Am F
is more common in phrygian. I don't really know the answer to this but it seems reasonable to me that people would tend to use different chord progressions in different modes because of the different opportunities each mode offers. I know know that I-iv-V is just as common in minor keys as I-IV-V is in major, but I believe that in "common practice" classical music the III(bIII) chord is much more common in minor keys than the iii chord is in major keys because III is the relative major. Again, I don't know much about common chord patterns in other modes, but it is mentioned in other answers.