Is Alice allowed to give private testimony to the jury/judge without Greg finding out about it?
NO, but there are mechanisms to safeguard Alice and her evidence from, for example, intimidation from Greg or people in the court's public gallery.
Article 6 of The European Convention on Human Rights (ECHR), which is incorporated in to law under the Human Rights Act 1998, requires everyone to be entitled to a fair trial, including, at Art6 (1)(3)(d) Greg being able...
...to examine or have examined witnesses against him ...
Note, it says "examine" not "directly question face-to-face across a busy courtroom".
Alice, more than likely, will be eligible for the assistance of Special Measures under the Youth Justice and Criminal Evidence Act 1999 due to...
...her age or incapacity section16 (colloquially referred to as a "vulnerable witness")
...her fear or distress about testifying section 17colloquially referred to as an "intimidated witness")
...she is a victim of a sexual offence section 22A
These Special Measures are:
Screening witness from accused. (available for vulnerable and intimidated witnesses): screens may be made available to shield the witness from the defendant, section 23
Evidence by live link (available for vulnerable and intimidated witnesses): a live link enables the witness to give evidence during the trial from outside the court through a visual link to the courtroom. The witness may be accommodated either within the court building or in a suitable location outside the court, section 24
Evidence given in private (available for some vulnerable and intimidated witnesses): exclusion from the court of members of the public and the press (except for one named person to represent the press) in cases involving sexual offences or intimidation by someone other than the accused. section 25 NB s.25(2) states that Greg and his legal team cannot be excluded.
removal of wigs and gowns by judges and barristers (available for vulnerable and intimidated witnesses at the Crown Court), ssection 26
Video recorded evidence in chief (available for vulnerable and intimidated witnesses): a visual recorded interview with a vulnerable or intimidated witness before the trial may be admitted by the court as the witness's evidence-in-chief, for adult complainants in sexual offence trials in the Crown Court. A visual recorded interview will be automatically admissible, upon application, unless this would not be in the interests of justice or would not maximise the quality of the complainant's evidence. There are also relaxed restrictions on a witness giving additional evidence in chief after the witness's visual recorded interview has been admitted section 27
Video recorded cross-examination or re-examination pre-trial. Visual recorded examination of the witness recorded at an earlier point in the process than the trial may be admitted by the court as the witness’s cross-examination and re-examination evidence in the Crown Court. This can only be applied for where there has been a s.27 direction for a visual recorded interview to be admitted as evidence and when a victim or a witness meets the vulnerable criteria. A visual recorded examination will be automatically admissible, upon application, unless this would not be in the interests of justice or would not maximise the quality of the complainant’s evidence, section 28
examination of the witness through an intermediary (only available for vulnerable witnesses): an intermediary may be appointed by the court to assist the witness to give their evidence at court. They can also provide communication assistance in the investigation stage - approval for admission of evidence so taken is then sought retrospectively. The intermediary is allowed to explain questions or answers so far as is necessary to enable them to be understood by the witness or the questioner but without changing the substance of the evidence section 29
aids to communication (only available for vulnerable witnesses): aids to communication may be permitted to enable a witness to give best evidence whether through a communicator or interpreter, or through a communication aid or technique, provided that the communication can be independently verified and understood by the court, section 30
(Sourced from the CPS, with some slight edits)
As a follow-up question, does Alice electing for some sort of "private" testimony weaken the case against Greg in any way?
NO - well it shouldn't but juries can be fickle and unpredictable at times - as the judge is required under section 32 of the 1999 Act to give a Warning the jury:
Where on a trial on indictment with a jury evidence has been given in accordance with a special measures direction, the judge must give the jury such warning (if any) as the judge considers necessary to ensure that the fact that the direction was given in relation to the witness does not prejudice the accused.