According to Reformed Theology if we need to willingly desire salvation to be saved do we contribute willingness, or receive it?
The concept of contributing willingness to be saved as opposed to receiving willingness to be saved is a new one to me, and I have been unable to find any specific reference to that. However, within reformed Protestant theology, there is a doctrine of Irresistible or Efficacious Grace that points to the office and work of the Holy Spirit in salvation. The following quote from the 1646 Westminster Confession of Faith may shed some light on this matter:
The doctrine of Irresistible or Efficacious Grace is set forth in the Westminster Confession of Faith: “All those whom God hath predestinated unto life, and those only, he is pleased, in his appointed and accepted time, effectually to call, by his word and Spirit, out of that state of sin and death in which they are by nature, to grace and salvation by Jesus Christ; enlightening their minds spiritually and savingly to understand the things of God; taking away their heart of stone, and giving unto them an heart of flesh; renewing their wills, and by his almighty power determining them to that which is good, and effectually drawing them to Jesus Christ; yet so as they come most freely, being made willing by his grace.” (Chapter X, Section 1)
This effectual call is of God's free and special grace alone, not from anything at all foreseen in man; who is altogether passive therein, until, being quickened and renewed by the Holy Spirit, he is thereby enabled to answer this call, and to embrace the grace offered and conveyed in it. (Chapter X, Section 2)
Another article on the Westminster Confession of Faith, partially quoted below, points to the Holy Spirit and the preaching of the Word as the means whereby God draws sinners unto salvation:
The Westminster Confession of Faith is a Reformed confession of faith. Drawn up by the 1646 Westminster Assembly as part of the Westminster Standards to be a confession of the Church of England, it became and remains the "subordinate standard" of doctrine in the Church of Scotland and has been influential within Presbyterian churches worldwide. https://en.wikipedia.org/wiki/Westminster_Confession_of_Faith
Chapters 10 through 18 describe various phases or aspects of salvation. The confession teaches that—by the Holy Spirit and the preaching of the word—God effectually calls the elect out of the state of sin and death toward faith in Jesus Christ and spiritual life. https://en.wikipedia.org/wiki/Westminster_Confession_of_Faith#Salvation
Nothing I have read suggests that the sinner actively contributes to their salvation by being willing. It is the power of the Holy Spirit that draws, by convicting the individual of their sinful state and then by that irresistible inner call. The following quote sums it up:
The doctrine of Irresistible or Efficacious Grace “asserts that the Holy Spirit never fails to bring to salvation those sinners whom He personally calls to Christ. He inevitably applies salvation to every sinner whom He intends to save, and it is His intention to save all the elect.” (Source: The Five Points of Calvinism by David N. Steele and Curtis C. Thomas (1963) p. 48)
Salvation is by Grace: “For it is by grace you have been saved, through faith – and this not from yourselves, it is the gift of God” (Romans 2:8).
Edit: I found an article that discusses the meaning of the term “effectual call”, as related to salvation, from Chapter X of the 1647 Westminster Confession of Faith. Here is a partial quote:
The effectual call is understood as God’s sovereign drawing of a sinner to salvation. The effectual call to a sinner so overwhelms his natural inclination to rebel that he willingly places faith in Jesus Christ. The apostle Paul refers to the effectual call when he writes, “It is God who works in you to will and to act in order to fulfill His good purpose” (Philippians 2:13). The necessity of the effectual call is emphasized in Jesus’ words, “No one can come to me unless the Father who sent me draws them” (John 6:44). https://www.gotquestions.org/effectual-calling-call.html
The rest of the article will shed light on how salvation is the work of God.