In English, "I saw my horse and dog" means "I saw my horse and my dog". But in Irish, "Chonaic mé mo chapall agus madra" does not mean "Chonaic mé mo chapall agus mo mhadra"; it means "Chonaic mé mo chapall agus madra éigin", doesn't it?

Can a possessive pronoun ever be used distributively in Irish, i.e to apply to two or more nouns?

An interesting question I cannot answer.
I'd think it can't.

Perhaps because the opposite is said to be possible, i.e. two possessors including a possessive adjective:
mo chapall féin agus Mhíchíl (my and Michael’s horse)
i m’fhochair féin agus Chormaic (in my and Cormac’s company)

Distributivity (if that's the right word) isn't quite common in Irish, even prepositions are usually repeated.

Spot on. I've found this to be true as well.

That is a very interesting observation. So "Chonaic mé mo chapall agus madra" might read like a poor stab at "I saw the horse that is mine and some dog's"!

On the other hand, "... my and Michael’s horse... ", if not actually ungrammatical English, is a construction that English speakers will jump through hoops to avoid, because we
powerfully prefer usage of "my" to stick to the distributive pattern " ...my X [,Y...[and Z]]...". Similarly "your","his","their" etc.

An bhfuil somplaí dúthchasacha agad dó sin? Tá cuma aisteach orthu, dar liom (déarfainn féin "mo chapall féin agus cionn Mhicheáil & i m'fhochair féin agus i bhfochair Chormaic").

_________________
Is fearr Gaeilg na Gaeltaċta ná Gaeilg ar biṫ eile
Agus is í Gaeilg Ġaoṫ Doḃair is binne

Léigh mé an dara cean in "Inflection and Silent Arguments in Irish" le J. McCloskey (https://people.ucsc.edu/~mcclosk/PDF/tromso.pdf). Ní bhfuair mé a fhoinse ("BO 069") sa téacs.

(I'd trust him to use sources by native speakers.
But on the other hand some of the examples in this paper are an-aisteach )

I see mistakes in his article:

(23) teach beag compórdach mo mhuitir -> should be "mo mhuintire"

p5 (in the tree) : arán laethúil na daoine -> should be "na ndaoine".

Maybe one could ask J. McCloskey himself!

_________________
Is fearr Gaeilg na Gaeltaċta ná Gaeilg ar biṫ eile
Agus is í Gaeilg Ġaoṫ Doḃair is binne

Chonaic mé mo chapall agus madra
"I saw my horse and a dog"

I think there would be ambiguity because Irish has no indefinite article, so Irish takes this sentence to mean "I saw my horse and a dog" and requires the repitition of the possessive pronoun for the meaning "I saw my [horse and dog]"

Chonaic mé mo chapall agus mo mhadra
"I saw my horse and a dog"

mo chapall féin agus Mhíchíl (my and Michael’s horse)
i m’fhochair féin agus Chormaic (in my and Cormac’s company)

From a purely logical perspective, these make sense to me because "Mhíchíl" and "Chormaic" are in the genitive.

mo chapall féin agus cionn Mhicheáil to me could logically indicate two separate horses, i.e., "my horse and Michael's one"

i m'fhochair féin agus i bhfochair Chormaic to me could logically indicate two different instances, for example, "X said Y in my presence, and also on another occasion in the presence of Cormac.

That's just a logical analysis. Actual usage is a different matter.

Sometimes Connacht irish diambiguates the homophonic ár mbád, bhur mbád and a mbád (all pronounced /ə mɑ:d/) by repeating the appropriate constrastive object pronoun after the noun, i.e., ár mbád => a mbád muidne, bhur mbád => a mbád sibhse, and a mbád => a mbád siadsan (parallel to Welsh possessive forms).

Could the ambiguity of the latter two sentences be avoided by the following? :

ár gcapall mise agus Mícheál (uilig)
inár bhfochair mé féin agus Cormac (uilig)

_________________
Múinteoir Gaeilge - Irish Teacher
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect, but I can also speak Ulster and Munster Irish with native-level pronunciation.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), Gaeilic Uladh (GU), Gaelainn na Mumhan (GM), agus Gaeilge an Chaighdeáin Oifigiúil (CO).

Interesting. How about with a comma? ár gcapall, mise agus Mícheál for example?