I will take a stab at answering, and there are several other posts that address this if you
search.
The short answer is academic/numismatic research, especially of the pattern of the location of finds - both
hoards and random finds.
The long answer...
For the later period it is easier. From the 290s on you get the first unambiguous
mint marks - with clues like
SIS for
Siscia and
LVG for
Lugdunum. But even these are not always clear and simple. The coins with T in the
mint mark (i.e. TT or P
T) were believed up to the early 20th century to be from
Tarraco (Tarragona,
Spain), but are now known to be from
Ticinum (Pavia,
Italy). It was careful analysis of the find locations of these coins - their distribution pattern - that finally settled this argument.
From the 270s - 290s there were marks that were less clear but that were at least distinguishable (i.e. P /
XXI, SXXI, XXIT). These also required distribution studies to
crack the code. But again things are not perfect. If you look a the coinage of
Aurelian (270 - 275) you get different interpretations of
mint location in the print version of RIC-V
part I, in Gobl's volume of
MIR, in Estiot's
work on the
Venera hoard, and in Estiot's online
work on an updated version of RIC-V
part I. Each newer
work can use the results of more research, more
hoard studies, etc. It is usually safest to go with the most recent
work - though only if it is by a knowledgeable scholar who has taken all previous
work into account. You have to be careful of recycled information. For example I could make a new website or book that simply recycles the 50-year old information in the old version of RIC-V
part I. It might be the newest
work in town, but it would be recycled out-of date information and people would be foolish to trust it over Estiot's
work.
The most difficult were the coins struck before there were any
mint marks, so before the 260s. There the main guide is
style, especially
bust style for
denarii and
antoniniani from the eastern mints like
Antioch. Of course in this period the rule is the main mints of
Rome, and for
part of the 1st century Lugdunum/Lyons.
Shawn