たがる and ほしがる: describing another person's wishes
Separate an observed desire to act from a desire for an object, and distinguish wanting from actually doing.
Use たがる for a desired action
自分で靴を履きたい directly expresses a wish to put on shoes by oneself. 娘は自分で靴を履きたがっています。 describes the speaker's observation that their daughter wants to put on her shoes by herself.
たがる attaches to the ます-stem: 見ます becomes 見たがる. A currently displayed desire can be expressed as 見たがっている. That statement alone does not establish that the person actually watched or saw anything.
Use ほしがる for wanting an object
子どもは新しい自転車を欲しがっています。 means that the child wants a new bicycle, as shown to the observer. Unlike 自転車が欲しい, which directly expresses a wish with 欲しい, this example marks the desired object with を.
たがる follows an action's stem, while 欲しがる takes the desired object. 自転車に乗りたがっている describes wanting to ride a bicycle; 自転車を欲しがっている describes wanting the bicycle itself.
Observation differs from direct quotation
弟は映画を見たがっています。 reports the observed wish of the speaker's younger brother to watch a movie. 弟は「映画を見たい」と言っています。 directly quotes him saying that he wants to watch a movie, so 見たい remains inside the quotation.
Do not turn this distinction into a rule that たい can never appear in any sentence about someone else. The key here is whether the speaker describes an observation or quotes the person's words. Neither たがっている nor 欲しがっている guarantees that the wish has been fulfilled.
Does 妹は自転車を欲しがっています。 establish that the speaker's younger sister bought a bicycle or is riding one now?
Show the answer and why
No. The sentence only describes her displayed desire for a bicycle. Purchase or riding would require separate information.