Hello,

First sentence:
Mi alparolis la kuraciston pri via problemo.

Second:
Mi priparolis vian problemon al la kuracisto.

Are they meaning the same?

Nope.

Let me first talk about alparoli because it is the simpler case.

According PIV the verb paroli has four meanings, of which three are intranstive (no direct object) and one transitive (takes a direct object). Without going into the details the meaning is "to speak, to discuss, to have a conversation".

When you put a preposition before a verb, the meaning gets shifted. Since the preposition al denotes movement towards something, as a prefix to paroli it shifts meaning to "to direct one's speech to someone, to address someone". For instance

• La fremdulo senĝene min alparolis : The stranger addressed me in a casual manner.
  
• Neniu el ili alparolis sian najbaron : None of them spoke to their(?) neighbour.

While you can express the theme of addressing with the preposition pri, I think more likely the sentence is going to have a cause why the person is addressed.

• La direktoro alparolis la personaron pri la ekonomia situacio. : The director let the personel know about the economic situation.
  
• La estro de la sekcio alparolis la kuraciston pro tio, ke… : The chief of the ward took up with the doctor the issue of…

(I hope some native English speaker checks my translations.)

about priparoli

Many verbs in Esperanto denote what could be described as simple actions:

• Mi plantas rozojn : I plant roses
  
• Li serĉas sian monujon : He is looking for his wallet.
  
• Mi nun respondos demandojn : I will answer any questions now.

These are simple actions in the sense that I plant only one type of plants, he is looking only for his wallet and I will do the answering now. Compare those with:

• Mi priplantas la ĝardenon : I plant the garden
  
• Li priserĉas la suspektaton : He performs a body inspection to the suspected.
  
• Mi prirespondos nur al Dio! : I will answer only to God!

Here the action of planting covers all kinds of plants, the search is not just for a single item and I am responsible for my actions only in front of God. So the prefix pri- gives the verb a broader or a shifted meaning.

There is also a question of case government (rekcio), i.e. what is the object or which preposition to use with the verb paroli.

All in all we get at least these alternatives:

• Mi parolis pri via problemo al la kuracisto. : I talked about your problem to the doctor.
  
• Mi parolis pri via problemo kun la kuracisto. : I discussed your problem with the doctor.
  
• Mi priparolis vian problemon kun la kuracisto. : I had a conversation about your problem with the doctor.

The first is more like I made a speech to the doctor. As I see the nuance difference between the second and third expression, is that the first (parolis pri) had more focus on the problem and I mainly did the talking while the second (priparolis) was a more wide discussion and more interactive.

WARNING!

Some people think whenever there is even a slight possibility of the preposition pri, i.e. "about" or "concerning", you can put pri as a prefix to the verb. This usually results in gibberish.

• Li donis al mi ringon → Li aldonis min je ringo → Li pridonis min je ringo.

Only the first sentence makes sense, the other two not.

Metsis:about priparoli

Many verbs in Esperanto denote what could be described as simple actions:

• Mi plantas rozojn : I plant roses
  
• Li serĉas sian monujon : He is looking for his wallet.
  
• Mi nun respondos demandojn : I will answer any questions now.

These are simple actions in the sense that I plant only one type of plants, he is looking only for his wallet and I will do the answering now. Compare those with:

• Mi priplantas la ĝardenon : I plant the garden
  
• Li priserĉas la suspektaton : He performs a body inspection to the suspected.
  
• Mi prirespondos nur al Dio! : I will answer only to God!

Here the action of planting covers all kinds of plants, the search is not just for a single item and I am responsible for my actions only in front of God. So the prefix pri- gives the verb a broader or a shifted meaning.

There is also a question of case government (rekcio), i.e. what is the object or which preposition to use with the verb paroli.

All in all we get at least these alternatives:

• Mi parolis pri via problemo al la kuracisto. : I talked about your problem to the doctor.
  
• Mi parolis pri via problemo kun la kuracisto. : I discussed your problem with the doctor.
  
• Mi priparolis vian problemon kun la kuracisto. : I had a conversation about your problem with the doctor.

The first is more like I made a speech to the doctor. As I see the nuance difference between the second and third expression, is that the first (parolis pri) had more focus on the problem and I mainly did the talking while the second (priparolis) was more wide discussion and more interactive.

WARNING!

Some people think whenever there is even a slight possibility of the preposition pri, i.e. "about" or "concerning", you can put pri as a prefix to the verb. This usually results in gibberish.

• Li donis al mi ringon → Li aldonis min je ringo → Li pridonis min je ringo.

Only the first sentence makes sense, the other two not.

Mi dankas vin. Via respondo estas plenega.