I understand that most clone effects on permanents do not target (i.e. the card text does not say "target").
Once the clone enters the battlefield, and I then announce what the clone is copying (i.e. I don't announce what I'm copying while the clone spell is on the stack), can opponents eliminate what I am copying to stop the copy effect?
In general, while a spell such as Clone is resolving, players don't have priority to cast spells or activate abilities (C.R. 117.2e, 117.1a-b). Thus, for example, after you decide that Clone will be a copy of a creature and announce which creature, players can't cast spells in an attempt to stop Clone from being a copy of that creature (or even from entering the battlefield); rather, by the time a player gets priority again, Clone will have already entered the battlefield (C.R. 608.3, 117.3b).
To be clear, ninjutsu is an activated ability which uses the stack (C.R. 702.49a, 602.1, 602.2). However, many of the same principles apply to Sakashima's Student entering the battlefield via ninjutsu as they do to Sakashima's Student resolving.
For example, players don't have priority to cast spells while an ability is resolving (C.R. 117.2e, 117.1a), so that my comment 2's second sentence applies accordingly to Sakashima's Student entering via ninjutsu or resolving as it does to Clone resolving. It likewise applies accordingly to Clone or Sakashima's Student entering the battlefield due to Aether Vial.