The suggestion to use two separate transformers is good one.

1) But only if they are of identical voltage could their outputs be placed
directly in parallel to increase the power (wattage) available.

Other wise one could feed into the other all the time and possibly cause
overheating. Even a small voltage difference could cause problems.
'Bell/chime' transformers are basically lightly designed for very
intermittent use so this would be very undesirable!

2) However it sounds as though it might be possible to have two separate
circuits with one transformer serving each chime; with the front door push
button completing the the circuit from both.

Although this also 'parallels' the transformers in a manner of speaking, it
does it through the two chime units (in series). The rated output voltage of
the two transformers however should still be be as close to identical as
possible. For eaxample if one transformer was, say 16 volts and the other
only 12 volts there would be, with arrangement #2 the difference of four (4)
volts '16 - 12 = 4, or a difference of twenty eight (28) 16 + 12 = 28. The
former might make the chime units at least 'buzz' or vibrate at a low level
all the time. The latter would operate them continuously!

There are some more complicated 'series' circuits, putting both transformer
outputs and chimes all in series with the door push that one could suggest.

Simplest if one is not electrically proficient, is get a bigger
transformer! But try the single one first; since it's a very intermittent
operation, how often IS your door bell rung daily? The existing transformer
may be adequate! the only trnsformer I've replaced was burnt out by being on
continuously for several months by a door push that 'stuck IN!